How did "muggins" come into use?

MacmillanDictionaryblog has a whole article about muggins which is an interesting read. About its origins it says that is definitely British and:

The word muggins apparently first occurred in the mid-nineteenth century, and probably came, for some obscure reason, from the surname Muggins, influenced by one of the meanings of mug – “a stupid person or a person who is easily fooled”.

It is a peculiar word that has the velleities of a personal pronoun:

We all know the list of English personal pronouns, but there’s one word that interests me because it seems to have the function of a personal pronoun but has very specific connotations. That word is muggins, which is defined in the Macmillan Dictionary as

“used for referring to yourself when you feel that you have allowed people to treat you in an unfair way”.

It is typically used, wryly or bitterly, in contexts where one person finds themselves doing a task, especially an unpleasant task, because others get them to do it or duck out of doing it themselves. Here are some examples:

• The first drive was to be done from Redditch to Knebworth with no support vehicle and muggins was to drive it.
• Ian has other priorities; Thom couldn’t organize his way out of a paper bag and isn’t keen anyway; so here is muggins doing all the work.

Etymonline adds some interesting details:

"fool, simpleton," 1855, of unknown origin, apparently from the surname and perhaps influenced by slang mug "dupe, fool" (1851; see mug (n.2)). It also was the name of simple card game (1855) and the word each player tried to call out before the other in the game when two cards matched. The name turns up frequently in humor magazines, "comic almanacks," etc. in 1840s and 1850s.

This thread, however, points out that Muggins may have been chosen randomly because it was such a common surname in the UK:

A muggins is a fool or a simpleton. It is probably based on the surname of a character in a Surtees story called 'Handley Cross', published in 1843. Presumably, therefore, Surtees just chose it as a typical, English-sounding surname, much as Buggins is. It cannot be an elaboration of 'mug', which also nowadays means fool/simpleton, because that slang meaning of the word did not appear until AFTER the story was published.

The origin is still unclear as suggested by the following article and its possible origin from “mug” appears to be inconsistent:

Muggins appears as a family name several times in eighteenth-century literary works — in particular by Tobias Smollett, John O’Keefe, Oliver Goldsmith and Charles Dibdin — often for an exciseman (a British official who collected excise duties and attempted to prevent smuggling) or some other person who is foolish or easily tricked. The earliest example in the Oxford English Dictionary is from 1703.

The OED also points out that Muggins belongs with a small set of supposed family names that indicate unsophisticated country attitudes, or a person easily imposed upon or lacking common sense. There’s Bumpkin (often generalised as country bumpkin), which probably comes from a Dutch word for a short, stumpy person, and Juggins, a nineteenth-century equivalent. Some writers have suggested that this last name comes from jug, which led them to argue that Muggins is indeed from mug.

And although mug, in its earlier sense of an unattractive face, is recorded from 1708 and so is contemporary with Muggins, mug meaning a stupid or gullible person is later, not being recorded in print until 1857.

(World Wide Words)

Instances of the surname 'Muggins' in published work from before 1800

A reading of the matches for muggins in a Google Books search for the word in books published from 1650 to 1800 strongly suggests that the word originated as a quasi-surname, and that it was (as the OED comment sited in user 66974's answer indicates) a "supposed family name" that "indicate[s] unsophisticated country attitudes, or a person easily imposed upon or lacking common sense."

Here, in chronological order, are the six Google Books search results involving occurrences of the surname Muggins in books published before 1800, together with the context in which each reference to Muggins appears.

From Tobias Smollett, The Adventures of Sir Launcelot Greaves (1760):

Farmer Stake, being first called to the bar, and sworn, touching the identity of Sir Launcelot Greaves and captain Crowe, declared, that the said Crowe had stopped him on the king's highway, and put him in bodily fear ; that he afterwards saw the said Crowe, with a pole or weapon, value three pence, breaking the king's peace, by committing assault and battery against the heads and shoulders of his majesty's liege subjects, Geoffroy Prickle, Hodge Dolt, Richard Bumpkin, Mary Fang, Catherine Rubble, and Margery Litter ; and that he saw Sir Launcelot Greaves, baronet, aiding, assisting, and comforting the said Crowe, contrary to the king's peace, and against the form of the statute.

Being asked if the defendant, when he stopped them, demanded their money, or threatened violence ; he answered, he could not say, inasmuch as the defendant spoke in an unknown language. Being interrogated if the defendant did not allow them to pass without using any violence, and if they did not pass unmolested ; the deponent replied in the affirmative : being required to tell for what reason they returned, and if the defendant Crowe was not assaulted before he began to use his weapon ; the deponent made no answer. The depositions of farmer Bumpkin and Muggins, as well as of Madge Litter and Mary Fang, were taken to much the same purpose ; and his worship earnestly exhorted them to an accommodation, observing, that they themselves were, in fact, the aggressors, and that Captain Crowe had done no more than exerted himself in his own defence.

Muggins is not before or again mentioned in Smollett's novel, but his appearance in close company with Farmer Bumpkin hints at a surname already laden with popular disesteem.

From Oliver Goldsmith, She Stoops to Conquer: or, The Mistakes of a Night (1773):

Tony. I can't stay, I tell you. The Three Pigeons expects me down every moment. There's some fun going forward.

Hardcastle. Ay ; the ale-house, the old place : I thought so.

Mrs. Hardcastle. A low, paltery set of fellows.

Tony. Not so low neither. There's Dick Muggins the exciseman, Jack Slang the horse-doctor, Little Aminidab that grinds the music box, and Tom Twist that spins the pewter platter.

And later:

Fourth [shabby] fellow [with punch and tobacco]. The genteel thing is the genteel thing at any time. If so be that a gentleman bees in a concatenation accordingly.

Third [shabby] fellow. I like the maxum of it, Master Muggins. What, tho' I am obliged to dance a bear, a man may be a gentleman for all that. May this be my poison if my bear ever dances but to the very genteelest of tunes. Water Parted, or the minuet in Ariadne.

The fourth shabby fellow is evidently Dick Muggins, the exciseman.

From "Intelligence Amorous and Polite," in The Covent Garden Magazine Or the Amorous Repository (December 1774):

Snob the Cobler, having been a short time at work, determines to go to the Alehouse ; which his wife Alice objecting to, the following dialogue ensues.

Snob. Why, you jade, what do you prate at?—Isn't Saint Monday, and are not the club waiting for me?

Alice. The club! Yes, indeed, you need squander your substance among a parcel of raggamuffins, 'till you b ring yourself, and your poor wife, to a jail.

Snob. Why, you impudent vixen, are Master Muggins, Joey Jenkings, and Gaffer Grumble, raggamuffins? Is the pimple-nos'd Exciseman a raggamuffin? or are little Hone the barber, Lawyer Pest, and Mr. Confusion, the surgeon, raggamuffins? But I don't know what's come to you, since you have had the credit of being a tradesman's wife.

From "Observation for June [1776]," in The Free-Masons' Calendar: or An Almanac, for the Year of Christ 1776 (1776):

[June] 13 | Tom o' Lincoln

14 | Dame Muggins

15 | Charles Vanity

F | Sir Poor Robin

From an untitled letter published in The Gentleman's Magazine (February 1789)"

I remember when a boy, hearing a man reproached for being ashamed of his name, and I was therefore taught to consider him as a contemptible fellow. Times are, however, now so much altered, that there is no derision attached to a weakness of that kind. When any person happens to inherit from his father what he thinks a vulgar or ill-sounding appellation, Potts or Watts, Pate or Bate, Huggins or Muggins, &c. he has only to apply for a sign manual, by virtue of which (on paying the fees), he may, if he pleases, call himself Howard, or Hastings, or Dudley, or Douglas, or Mordaunt, or Montagu. I think I have, within these few years, counted in the London Gazette upwards of an hundred of the most obscure names exchanged in this manner for others, which have struck the adopters as illustrious and musical; and what I particularly remarked was, that a considerable number of these new Christians were tradesmen of Leeds, Sheffield, and other manufacturing towns, who, I presume, expect by this means a sort of new birth or baptism of gentility.

From an untitled submission published in The By-stander; Or, Universal Weekly Expositor (1790):

Standing in the field the other day, I heard the following conversation between two beggars.

...

'Aye, aye,' said the other, 'I remember it well enough : but, now you talk of that, how came your leg so plump and jolly? I hopes as how you was not afraid of a judgment upon your sweating it don?' "A judgment,' said the other, 'I'll tell you all how and about it. You see I was always a clever fellow at transmogrifying a limb, so I tied up my peg harder and harder every day, just as they do with the women in some outlandish country, where they make them wear iron shoes ; and so you see my leg was at last as thin as Tom Muggins's, the notamy in surgeon's hall.

My impression is that "Tom Muggins" was a nickname given to a skeleton kept on display in "surgeon's hall."

'Muggin[s]' as a Scottish term

In addition to the preceding six instances of Muggins as a proper name, there is one instance of muggins as an uncapitalized common word. From "Woo'd and Married and A'" in A Collection of Original Scots Songs, Poems, &c., by Various Hands (1772):

Had they kent her but as weel as I did, / Their errand it wad ha' been sma', / She neither kens carding nor spinning, / Nor baking nor brewing at a'; / But a' is to gang fine laced, / And gang white-finger'd and bra'; / I'm afraid she's quite out o' the fashion, / So line up her muggins wi' straw.

Chambers Scots Dictionary (1911) offers this (probably) relevant definition of muggin:

Muggin, n. a long, footless stocking. Cf. Moggan.

Chambers's entry for moggan is more detailed:

Moggan, Moggen, Moggin, n. a stocking ; a footless stocking ; a stocking used as a purse ; a long stocking-like sleeve for a woman's arm ; in pl. the legs.

So, in the song, muggins may be legs, stockings, stocking-like sleeves, or stockings used as purses.

John Jamieson, An Etymological Dictionary of the Scottish Language (1808) has a similar entry for moggans:

MOGGANS, s. pl. 1. Long sleeves for a woman's arms, wrought like stockings, S.B. [Citations omitted] 2. Hose, without feet, Aberd[een]. ...

'Muggins' as slang for a dolt or fool

Although the Scottish word muggin[s] is interesting, I don't think it is likely to have significantly influenced the emergence and popularization of Muggins as a stereotypical hick-from-the-sticks surname. That popularization may be due in large part to its use by two very popular authors of the latter half of the eighteenth century—Tobias Smollett (writing in 1760) and Oliver Goldsmith (writing in 1773)—although they may have picked up the name from earlier informal usage. Both use Muggins not for a major character but, in passing, for a minor character who appears with other characters who have droll surnames such as Prickle, Dolt, Bumpkin, Fang, Rubble, Litter, Slang, Aminidab, and Twist (long before Oliver Twist). In the nineteenth century, several authors gave characters named Muggins considerably bigger roles in their fiction—for example, Peter Muggins and family in George Reynolds, Pickwick Abroad; or, The Tour in France (1839); Bill Muggins in Charles Selby, Maximums and Speciments of William Muggins, Natural Philosopher and Citizen of the World (1841), Mrs. Muggins in George James, The Commission: Or, De Lunatico Inquirendo (1843), Giles Muggins in Les Anglais Pour Rire; or Parisian Adventures in The Metropolitan Magazine (July and August 1846), and Muggins in All's a Delusion: A Comedy (1847).

Some of these Mugginses are rustics, some are big-city working-class people who sound like Sam Weller, one is a landlady, and one a soldier. But all are English or Irish, and virtually all are characters in comedies.

The earliest entry for muggins as a slang term that I've been able to find is in J.S. Farmer & W.E. Henley, Slang and Its Analogues Past and Present, volume 4 (1896):

Muggins, subs. (common).—1. A fool. ... 2. (common).—a borough-magnate ; a local leader.

But this is substantially later than the earliest dictionary or glossary occurrence of mug in the slang sense of "fool" that I found. From George Matsell, Vocabulum: Or, The Rogue's Lexicon (1859):

MUG. The face ; a simple fellow.

And Farmer & Henley cites an even earlier instance of the usage from Henry Mahew, London Labour and and the London Poor, volume 3 (1851), although it isn't clear that the Mayhew instance involves using mug to characterize someone as (as Farmer & Henley puts it) "a dolt" or "a raw, or clumsy hand" and not as something more closely related to mugging in the sense of "paint or make-up":

We ["Ethiopian serenaders"] sometimes have a greenhorn wants to go out pitching with us—a mug we calls them ; and there's a chap of the name of 'Sparrow-back', as we called him, because he always wore a bob-tailed coat, and was a rare swell ; and he wanted to go out with us, and we told him that he must have his head shaved first, and Tom held him down while I shaved him, and I took every bit of hair off him. Then he underwent the operation of mugging him up with oil-colour paint, black, and not forgetting the lips, red.

In any case, muggins for "fool" is undoubtedly influenced by the tradition of comic Mugginses in British literature from (at least) 1760 forward; but mug for "mouth" or "face" and subsequently for "fool" seems likely to be the direct source, rather than being a shortening of muggins at some still earlier (but unrecorded) date.