How did spitballing originate

Early Spitballs

The 'spit ball' (open compound) appears as early as 1794. At that time, the reference was to an object I surmise, on the shaky basis of slightly later evidence, was used as an aid in blackening shoes:

ISAAC BULL
Has very lately receiv'd a supply of DRUGS and MEDICINES: Also, Painters Colours, and almost every kind of Grocery.... The following are a part of his assortment, viz.
....
Rosin & Spirits Turpentine;
Spit Ball,
Sweet scented Wash Ball,
Smelling Bottles with pungent Salts,
Tooth Brushes,
[etc.]

Hartford Courant (Hartford, Connecticut), 30 Jun 1794 (paywalled; emphasis mine).

The slightly later evidence basing my surmise was this from the Hartford Courant of 13 Jul 1801 (paywalled):

SHOE STORE
WILLIAM CHADWICK,
Takes this method to inform his customers, and the public in general, that he has received from New-York a pretty general assortment of Morocco Leather of different kinds, for sale, ... a few doz. Spit Ball for Shoe Blacking, of a superior quality.

The foregoing attestations are of tangential interest here primarily because the object described as a 'spit ball' is not mentioned in other reference works. Also, the late 18th century attestation is much earlier than the 'spitball' (frequently a closed or hyphenated compound) defined in OED as a "spittle-ball [ball of chewed paper wet with saliva], esp. one thrown as a missile by a schoolchild".

This first sense of 'spit ball' appears nowhere except in the Hartford Courant, and there doesn't recur after 1802.

The OED's first attestation of the "missile" sense from Knickerbocker, vol. 27, 1846, can be antedated 7 years to 1839:

...not covered as they [schoolhouse walls] are now, with the first engines of war, which the little urchins manufacture from the spare leaves of their books, in other words, spit-balls, daubs of ink, and even obscene writing, and representations that would make a parent of any sensibility shudder at the thought of sending a child to such a place.

The Common School Journal. For the Year 1839, Vol. I, edited by Horace Mann. Emphasis mine.

26 Nov 2020 Addition 1: Thanks to DavePhD (a comment on this answer), I can cite an earlier antedating to the journal entry of 16 Nov 1837 from Journals and miscellaneous notebooks of Ralph Waldo Emerson: Volume V 1835 – 1838 (1965, Merton M. Sealts, Jr., ed.):

I am glad to have poor Puckler Muskau however so butchered as he is in the Quarterly with all kinds of rotten eggs & spit balls pelted...

Wet Ball Baseball

A sense of 'spitball' from the enormously popular North American sport, baseball, was common in the early 1900s and continues to be widely used:

A ball moistened on one side with saliva or sweat before pitching, so that it acquires a swerve. (Illegal in the official game [as of 1920, excepting pitchers 'grandfathered' in through 1934].)

OED sense 2. Earliest attestation 1905.

US newspapers adopted the term in this sense from its use by baseball players, and have employed it frequently since at least 1903, at which time it prevailed against a competing term, 'wet ball', in use since before 1876:

...Simmons out at first; Abodie safe on muff of wet ball by Sullivan; J. Gleason and Galvin out on fly to Raja and Clinton.

Memphis Daily Appeal (Memphis, Tennessee), 16 May 1876 (paywalled).

The players call it the "spit ball," and the umpires call it the "wet ball." The name makes but little difference, however, as it amounts to the same thing.

The Salt Lake Herald (Salt Lake City, Utah) 20 Jul 1903 (paywalled).

Going head to head against the colorful 'spitball', that "first engine of war", it was inevitable that the bland, flavorless competitor 'wet ball' would fail.

Miscellaneous Uses

In the years between the appearance of 'spitball' in the "missile" sense sometime prior to 1839, and its later use in the "wet ball" baseball sense, a number of semi-literal and figurative uses began to appear. Many of these refer to political and legal 'mud-slinging'. Such uses continue to this day. Here are some early examples from the popular press:

MR M. F. DICKINSON, JR.,
spoke of the character of the School Board. He said that more time was lost in wrangling and throwing spit-balls, in personalities, and in waiting for quorums than in any body he ever say.

Boston Post (Boston, Massachusetts), 02 Feb 1874 (paywalled).

Hard money men, strip for the fight and stamp out the inflationists! If they attempt to annihilate the credit of the country with their rag money spit balls, return their fire with a charge of specie payment arguments that will riddle their paper boat and strand them in the sea of Repudation, where they would founder the Republic.

Carlisle Weekly Herald (Carlisle, Pennsylvania), 23 Sep 1875 (paywalled).

Up to it swells with importance a portly citizen, who imagines that upon the deposit of his vote hinges the revolution of the world. And he puts it in, and the inside officers, if it is right, put it in the box and count it with a devotion worthy of a camp meeting; but if it is not right they chew it up and make spit-balls out of it, and, in order that the effort of the citizen should not be lost, drop two in its place in the box, and go out and blow about the terrible iniquities of the Democratic party.

The Times (Philadelphia, Pennsylvania), 28 Jun 1876 (paywalled).

Such uses are simple metaphors leveraging the "missile" sense. 'Spitball' signifies anything petty and offensive 'thrown' or 'blown', in some sense, at others.

Later Senses

Along with continued metaphorical use in the sense of "an insult, off-color remark; verbal abuse", 'spitball' enjoyed brief, localized slang use designating a particular type of "hand grenade", as recorded by Farrow in A Dictionary of Military Terms, 1918:

Spit-ball Grenade. — A grenade about the size of a baseball, equipped with an attachment which will detonate it only when it is thrown as a spit ball. The grenade leaves the hand in the same way that the so-called "spit" curved ball is delivered by a baseball pitcher.

By 1925, the specialized sense designating a particular type of hand grenade was recorded as a general reference to "hand grenade":

1925 E. Fraser & J. Gibbons Soldier & Sailor Words 266 Spit ball, hand-grenade. (U.S. Army.)

From the OED quotation collection.

It can safely be assumed that other localized and specialized senses of 'spitball' have been used, however restricted or brief such uses have been. By their very nature, spitballs paper the US cultural walls.

The Derivative Verbs

Most of the senses of 'spitball' given in the foregoing account enjoyed use as verbs. The exception is the very early, 18th century, sense of "an object employed as an aid in blackening shoes". Otherwise, for example, and without belaboring the point, 'spitballing' described the action of hurling paper missiles, throwing a wet baseball, and so on.

However, one later sense of 'spitballing', the sense of "to throw out suggestions for discussion" (OED), seems to have developed without much (if any) precursor figurative use of 'spitball' in the more neutral, general sense of "idea, suggestion, topic". The use as a verb certainly derives from earlier use of 'spitball' and 'spitballing' with reference to saliva-soaked paper missiles, then wet baseballs, but in this use the sense of "missile" has weakened to include anything being 'tossed around', speaking figuratively.

Were it not for a couple of early contradictory attestations, the later, weakened and derivative verbal use of 'spitballing' (and 'spitballed') in the general sense of "bringing up ideas and topics for discussion" would seem to have originated from repeated use of 'spitballing' in the 'trademark phrase' of a particular syndicated columnist. That columnist, Jack Crosby, a music, radio, television, drama, theater, movie et al. critic, used this 'trademark phrase' in widely reprinted columns at least once a year between 1950 (the earliest use I could find of 'spitballing' in this derivative sense) and the 1960s. His phrase took the form of direct address to a perhaps imaginary friend named "Mannie". Examples include these:

   The fact is, Mannie, a girl has got a right to enjoy herself. It's later than she thinks. She'll be out of this world soon enough and she'll have to answer for her indiscretions to higher authority than Irving Berlin. Which gives me a great idea for a song: "Will You Love Me in Hell as You Do in Connecticut?" ("We got a date at the Stygian gate. Now don't be late. We gotta be there when the bell starts knelling.")
   I'm just spitballing there, Mannie, but you get the idea. Now, if we can just get the Andrews Sisters...that's as far as I've gone.

From a column titled "Radio In Review", The Daily Times (Salisbury, Maryland), 04 Aug 1950 (paywalled).

   What's the matter with television, boys?
   "She was a vision
   "On television . . .
   I'm just spitballing here, Mannie, but you can see right away it can't miss.

From a column titled "Radio and Television", Lansing State Journal, (Lansing, Michigan), 03 Nov 1950 (paywalled, emphasis mine).

   Let's say you're sitting with a couple of actresses, an actor and a director watching a television play in your own living room. In this play — I'm just spitballing here, Mannie — a beautiful wanton is plotting with her lover to kill her husband. So help me God, the first bit of criticism you will hear — from one of the actresses — will be:
   "Where do you suppose Marcia got that hair-do — from her psychiatrist?"

From a column titled "Radio and Television", Lansing State Journal (Lansing, Michigan), 06 Jul 1953 (paywalled, emphasis mine).

The drumbeat of Crosby's trademark phrase in his columns, and its near-singularity in the sense ("tossing out ideas") continues into the 1960s. Were it not for William Morris's thoughtful intervention (as follows) with other information in 1953 and 1954, Crosby's phrase might be taken as the original use of this derivative verbal sense.

** — William Morris**
Words, Wit & Wisdom
One of my favorite — if somewhat conservative — correspondents, G. M. Staplin of Utica (N.Y.), has written again about something which I am sure has vexed all of us at one time or another.
   "One of my pet peeves is the way English is being mistreated on the radio, where of all places it would seem that correct grammar should be observed. Employees of local stations may sometimes be excused if they occasionally "slip," but to hear "Here is how" and "Here is why" from highly paid and theoretically, at least, highly trained announcers grates on my sensitive ears. To my mind, the tragedy of it is that such expressions as these, which are heard time and again in the course of a day, may eventually be recognized as proper diction. However, the script and not the reader must take the blame."
   ANSWER: Amen! ....
....
But this highfalutin talk is commonplace in the hyperthyroid world of Advertising Row where one no longer "thinks out loud" — he "flies blind for a minute here, just spitballing, so to speak." And I should cheerfully argue that copywriters exposed to this fantastic jargon day in and day out cannot be blamed too much for locutions as comparatively sanctioned by common usage as "Here is how..." and "Here is why..."

Reno Gazette-Journal (Reno, Nevada), 15 Dec 1953 (paywalled, emphasis mine).

Words, Wit & Wisdom
By William Morris The Scarsdale Galahad — II
   Just how does a village of 15,000 merit being chosen as the place where "English is spoken best in the United States" — a nation of over 160,000,000? Let Max Sherover, head of Linguaphone Institute, who made this astonishing statement and caused so many of my readers to explain it for you.
   "Scarsdale," he says, "proportionally has more advertising executives, Wall Street tycoons, business leaders, college graduates and folks with greater general information than any other town in the land." Elsewhere he notes that Scarsdale residents "have the highest per capita income in the nation."
   Well, let's look at the evidence. First among the groups which he claims contribute to Scarsdale's alleged ascendancy in spoken English are "advertising executives". These are the people responsible for such gems of literate expression as: X-Bar-X Cigarettes Travel the Smoke Further; ABC Rum Contains Less Calories; and This Really is Real Coffee. What's more, the daily speech of many ad men is studded with expressions like "spitballing" and "flying blind" — both meaning "improvising" —  and "talking off the top of my head" for "thinking out loud." All of this may indeed be colorful and not without a certain charm and vigor — but it hardly qualifies as "best spoken English."

The San Bernardino County Sun (San Bernardino, California) 29 Jan 1954 (paywalled, emphasis mine).

From these instances, it is apparent that a very well-informed and expert observer of English usage, William Morris, attributes the use of 'spitballing', when it means verbal "improvising" (closely akin to OED sense 2, "throwing out topics for discussion"), to use in advertising jargon.

26 Nov 2020 Addition 2: Thanks to a suggestion by D Krueger (in a comment on this answer) and temporary access to The New Yorker archives (paywalled), I was able to locate an early use of 'spitballing', in the sense of "throwing out ideas", in a story by S.J. Perelman published 16 Jan 1937:

Mind you, this ain't dialogue — I'm just spitballing.

This use, the earliest in the sense of "throwing out ideas" that I've found, is given in a context of Hollywood movie-making story pitches, and so suggests that origin of the sense, rather than advertising jargon. Still, my reasoning in 2017 regarding development of the sense from the baseball sense of spitball pitch to an advertising pitch transfers readily to development from the baseball pitch to the Hollywood story pitch.

The Pitch

As mentioned under the previous heading, The Derivative Verbs, 'spitballing' in the sense used in advertising jargon, 'to improvise; to conceive, propose and discuss ideas or topics', seems to have developed without the precursor use of 'spitball' as a noun in the sense of 'idea, topic'. While my not having found use in that sense may simply represent a gap in the evidence examined by me, and the noun may in fact have been so used, an alternate development route seems likely.

The likely route for development of the derivative verb 'spitballing' in advertising jargon, without precursor use of the noun 'spitball' in the sense of 'idea, topic', is logically straightforward via the 'pitch'.

The verb 'to pitch' has this sense when applied to advertisement:

To try to sell (merchandise) by persuasion, esp. by drawing attention to specific attractions, advantages, etc. Now usually fig.: to promote the acceptance of (an idea, proposal, project, etc.) in this manner.... Also: to approach (someone) in order to obtain support for an idea, project, etc.

OED, pitch, v.², sense 15d. First attested 1926.

The corresponding noun sense of 'pitch' is this:

Orig. slang. Speech or other behaviour designed to persuade, influence, or cajole, esp. in order to sell goods or promote an idea; patter, spiel; an instance of this.

OED, pitch, n.², sense 9b. First attested 1876.

Although I have no 'smoking gun' of evidence establishing a connection between 'pitching' advertisements and 'spitballing' (that is, 'improvising and proposing') ideas for advertisements, the parallelism of pitching spitballs in baseball and 'spitballing' as a type of advertising 'pitch' recommends itself. Additionally, terminological chronology does not contradict adoption in advertising from use in baseball; 'pitch' is used to describe the process of selling in advertising as early as 1882:

After thinking about it for a time, the St. Louis Globe-Democrat come to the conclusion that "if Mrs. Langtry can't act she can, at least, work the scandal racket up to the highest advertising pitch."

The Topeka Daily Capital (Topeka, Kansas), 15 Dec 1882 (paywalled).

Even supposing that 1882 collocation is not a pun leveraging both the "comparative degree" and the "persuasive speech" senses of 'pitch' as a noun, and the intended meaning of 'pitch' as used is only "comparative degree", the meaning of a subsequent collocation in 1899 also suggests the "persuasive speech" sense of 'pitch':

   Much advertising fails reaching its highest effectiveness because the parts of the plan on which is done do not fit into one another.
   The mere insertion of advertising in newspapers and magazines is not enough, no matter how good the advertising is.
   The entire business must be kept up to the advertising pitch.

Marketing Communications, v.29, 1899.

To provide a logical source as suggested, however, the baseball sense of 'spitball pitch' and 'spitballing' would need to have preceded the adoption of 'spitballing' in advertising jargon to describe a type of advertising pitch. As nearly as I can determine, use of 'spitball' to name a type of baseball pitch sometime before 1903 (see above), and use of 'spitballing' in 1906 (see below) to describe the action of spitball pitches, does precede the adoption of 'spitballing' in advertising jargon sometime before 1950.

...for several innings we have been trying with such skill as the years have left with us to connect our willows with the twisting sphere that you have been bending and shooting and curving and dropping and spit-balling by us.

(Scranton Republican, Scranton, Pennsylvania,27 Apr 1906, paywalled)

The full expression is something like "Let's spitball it and see if it sticks." A spitball was typically thrown against a wall or ceiling and the challenge was to get it to stick rather than fall off. It's the "get it to stick" part that is not spoken but alluded to: Try different "crazy" ideas and see if any "stick" and seem to have real potential.

(Yeah, a "spitball" (sometimes called a "spitwad") is a piece of paper, maybe an inch and a half square, that has been wadded up and chewed until it's gummy. (If you want it really gummy get some snot mixed in.) I'm now remembering that the "expert" spitballers could blow the wad out of their mouths, by rolling their tongue and blowing through it.)

Spitball:

To speculate; propose conclusions or possibilities : Well, I'm just spit-balling/ You're just spit-balling (1955+)

• [second noun and verb senses fr the mischievous schoolboy's vice of throwing bits of paper soaked in saliva; verb sense fr the notion of tossing such spitballs more or less idly]

(The Dictionary of American Slang, Fourth Edition)

Spitballing in the schoolboy sense is much older:

• 1846 in the schoolboy sense, "bit of paper chewed and rounded as a missile.

(Etymonline)

According to the following source the term, in its fugurative sense, is more often used in politics:

• It appears that the expression is one that’s been quite popular, especially in the realm of politics.

Andrew Malcolm from the Los Angeles Times on June 15, 2010.

• For a man who has repeatedly claimed to be “fully engaged since Day 1,” and who repeated that claim last night, Obama gave every impression of still being in the spitballing stage of crisis management.

Patrick Sauer wrote an article published by the Huffington Post:

• Angry folks are practically calling for a public guillotining of a Wall St. fat cat or two. Ironically, and I’m just spitballing here, that bloody spectacle would do boffo business on pay-per-view and easily cover the next TARP giveaway..

(idiomation.wordpress.com)

According to William Safire, etymology columnist for the New York Times, in his 1981 book On Language:

Because an old-time baseball pitcher never knew which way his spitball would break, the verb to spitball now means “to speculate."

Of course, Safire could just be spitballing.