I looked up some history of this word, first at the Word Detective:
It seems that this use of "dibs" is not nearly as old as one might have thought. The first citation for the "it's mine" sense of "dibs," in fact, comes only from 1932, although "dibs" was used as slang for "money" as early as 1807. "Dibs" was an abbreviation of "dibstones," a game similar to jacks played with "dibs," which were sheep knuckle bones. It is unclear exactly how "dibs" came to be a child's way of staking a claim, but presumably it made sense if you knew the rules of the game of "dibs."
...a fascinating list of the British equivalents. A child in Southern England who spots the one cookie left on the plate might exclaim "Bags it" or "Baggsy," whereupon by the sacred code of children the prize is hers. Her London counterpart might say "Squits," and still further north a child would say "Foggy," "Furry" or "Firsy." Other words which seem to work as well include "Barley," "Bollars," "Jigs" and, in Scotland, "Chaps" or "Chucks." We will probably never know where these words came from...
Michael Quinion's World Wide Words states, in part:
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.