The Oxford English Dictionary lists the following meanings of the word dibs:
—A game played by children (also called dibbs, or dibstones) with pebbles or the knucklebones of sheep; also the name of the play pieces (late 1600s)
—A children’s word to make first claim on something (1907)
—A counter used in card games as a substitute for money
—A slang term for money (1808)
It also meant a depression in the ground, perhaps 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.”
Dibstones is known to have been played since 1693 as it was recorded in the English philosopher Jock Locke’s Some Thoughts Concerning Education, published that year.
“Dib” originally was a verb meaning “to dab” or “to pat.” The game of dibstones closely resembled the game of jacks, often involving tossing up small objects and catching them on the backs of hands. In other words, “putting” them somewhere. Other forms of dibstones were similar to the game of marbles.
As for privilege, it’s likely that the game allowed a player to gain privileges over their opponents if the dibs went a certain way. Or perhaps dibs was influenced by dubs, a shortened form of double that is used in the game of marbles as an exclamation to declare one’s right to two marbles knocked outside the ring of play. If dibs came to be used in a similar way, it is possible that its meaning broadened during the 1900s to convey the more general sense of “rights” or “claim” that it possesses today.
As to the “money” sense, 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.”
Dibs and dibstones originated in England but all of the early citations of “first dibs” are from the USA, beginning in the early 1900s. The Wisconsin Home and Farm School Association was a charitable group that aimed to provide housing and education for homeless boys. In 1907 they published a pamphlet, including a verse which ends with:
“I got first dibs
On the baking pan.”
I never used “first dibs” as a child, but with my siblings rarely around, I didn’t need to. I always got to lick the icing bowl.