What is the origin of "dibs"?
According to World Wide Words, its usage appears earlier than 1932. But the usage in the first part of the 19th century may refer to a different meaning.
What we do know is that this expression is first recorded in print, in American Speech, as late as 1932. It comes into existence seemingly fully formed, with no obvious links to any previous meaning of the word. That’s hardly likely, of course.
Most writers seize on what seems to be the most relevant older use of dib as a word connected with childhood. This refers to an ancient and very common game known by dozens of other names (jacks, fivestones, knucklebones, hucklebones; pentalithia in classical Rome), though the name dibs is recorded only from the early part of the eighteenth century. Here’s a late reference from Thomas Hardy’s Jude The Obscure of 1895: “Why when I and my poor man were married we thought no more o’t than of a game o’ dibs!”. It seems to be an abbreviation of an even older term, dibstones, a name whose origin is obscure to the point of terminal murkiness. The problem is that we have no idea how a word for a game in Britain turned into an American expression claiming priority (British children would often use bags in this situation, a term derived from public school slang). Another sense of the word which is sometimes put in evidence is the slang one meaning money. Here’s H G Wells, in The War in the Air: “He thought the whole duty of man was to be smarter than his fellows, get his hands, as he put it, ‘on the dibs,’ and have a good time”.
There are various other meanings of dib, as both noun and verb, which has had a muddled history in which dab and dap feature strongly as variant forms. But none of these have any obvious link to the word in the sense you’re asking about. As an example, in older northern English dialects it meant a depression in the ground, possibly a variant of dip, as here in John Galt’s The Annals of the Parish of 1821: “The spring was slow of coming, and cold and wet when it did come; the dibs were full, the roads foul, and the ground that should have been dry at the seed-time, was as claggy as clay, and clung to the harrow”.
Yet another suggestion is that the word is a modified abbreviation of division or divide. This neatly circumvents the problems with provenance, and fits the model of many children’s slang terms of this and earlier periods. But I’ve not come across any evidence for it.
J.E. Lighter, The Random House Historical Dictionary of American Slang (1994) puts the first occurrences of dibs in print at 1807 (used in the sense of money), 1827 (used in the sense of a portion or share), and 1932 (used in the sense of a first claim):
dib n. {prob. fr. dibstones, a type of child's jacks} 1.a. pl. money. 1807 Port Folio (June 6)357: You must put some more cash in your pocket./Make Nunky surrender his dibbs. 1821 Real Life in London II 57: The dibs are in tune—There is plenty of money. [Other citations omitted.] b. a dollar [Citations omitted.] 2.a. a share; portion. 1829 in J. Farmer Musa Pedestris 107: If you'd share the swag, or have one dib...{i.e.} the least share. 1859 Matsell Vocab. 25: Dib. Portion or share. b. pl. esp. Juve[nile] a first claim on an item. 1932 A[merican] S[peech] VII 401: Dibs on that magazine when you're through. [Other citations omitted.]
The fuller context of some of the citations in Lighter's discussion of dibs is as follows. From "The Song of George Barnwell," in The Port Folio (June 6, 1807):
Then said Millwood, whose cruel heart's core,/'Twas so cruel that nothing could shock it,/If you mean to come home any more,/You must put some more cash in your pocket./Make Nunky surrender his dibbs,/Wipe his pate with a pair of lead towels,/Or stick a knife into his ribs—/I warrant he'll then show more bowels.
From Pierce Egan, Real Life in London; or The rambles and Adventures of Bob Tallyho, Esq. (1821):
“All right my boy!” exclaimed Merrywell “bring your bellows ["snuff"],in good order, and don't be afraid' of your bread basket ["stomach"].“ The dibs are in tune. A ball of fire ["glass of brandy"], a dose of daffy ["Daffy's Elixir," a patent medicine introduced in the 1650s], or a blow out of black strap ["drink of gin or rum mixed with molasses"], will set the blue devils at defiance, give a spur to harmony, and set the spirits a jogging."
From "On the Prigging Lay" (1829), in John Farmer, Musa Pedestris: Three Centuries of Canting Songs and Slang (1896):
When twelve bells chimed, the prigs ["thieves"] returned,/And rapped at the ken ["house"] of Uncle ——:/"Uncle, open the door of your crib/If you'd share the swag ["plunder"], or have one dib./Quickly draw the bolt of your ken,/Or we'll not shell out a mag ["half-penny"], old ——."
From George Matsell, Vocabulum: Or, The Rogue's Lexicon (1859) [a glossary entry]:
DIB. Portion or share.
From these citations, it appears that the meaning of dibbs/dibs evolved in slow but orderly fashion from dibstones (the children's plaything) to money to a portion or share in some benefit or distribution to a first registered claim (as of right) to possess something.
Since this subject also came up in relation to Royal Army/Navy slang, I’ll go ahead and throw in a late answer for OP’s original question:
Given this information, why does Etymonline trace its origin to this low point in 1932 when it was actually in use over a century earlier?
First, in general, because Etymonline is run by a nice enough bloke but it’s just Mr Harper, he makes plenty of mistakes, and you should not treat it as a reliable source any more than a Wikipedia or Wiktionary article. Sure, most of the entries are fine, but he’s mostly just collating sources without checking or stating them. Take anything listed with an ounce of salt and try to figure out what his original source was, which is usually going to be OED entries.
Second, in this particular case, the reason he gives the date 1932 is because that’s the earliest citation for that sense of “dib” given by the OED:
1932 [Leonard W. Merryweather “The Argot of an Orphan’s Home”] Amer[ican] Speech [August Volume] 7 [Number 6 Page] 401[:]
Dibs, interj[ection], an interjection giving option on first chance or place. “Dibs on that magazine when you’re through.” “Dibs on going with the team if there’s room.”
Of course, that article is reporting something that’s already established slang, in this case at the unnamed “campus of a large home for orphans” where Mr Merryweather “recently... worked for a year” in “the presence of over a thousand inmates, with adults in the ratio of about one to ten.” He reports that the kids quickly homogenized their pronunciations and expressions to conform with the group, regardless of background, although their mothers (?!), “many of whom are employed on the campus, retain their southern, midwestern, New England, Canadian, and even English pronunciation[s] through years of residence at the home.”
Third, I don’t really agree with the checked answer by @user66974. Games of knucklebones or jacks—where the counters are quickly claimed and kept by successful players—lend themselves obviously and inevitably to possessive claims of “MINE!” w/r/t successful play. I don’t think the OED or other dictionaries are out on a limb considering that sense to be derivative here.