What is the origin of "dibs"?

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.

From the excellent phrases.co.uk: https://www.phrases.org.uk/meanings/first-dibs.html

It also seems likely that the game dibs, a.k.a. dibstones, which involves the claiming of the dibstones by calling out 'dibs' is how 'first dibs' originated.

Dibstones is known to have been played since 1693 as it was recorded in the English philosopher Jock Locke's Some Thoughts Concerning Education, which was published that year:

So, it's earlier than 1693.