"Uncle" is called by one child for another to submit or cry for mercy — which appears variously as say uncle!, cry uncle! or holler uncle! — is first recorded in print in the U.S. early in the 20th century. The Oxford English Dictionary’s first example is from 1918, but I’ve found an instance in an advertisement in the Modesto News of California, dated 1912: "This Time it is ‘Martie’ Graves and Don Johns who made them say ‘Uncle.’"

The speculations are ingenious: one from American Speech in 1980 was "Uncle in this expression is surely a folk etymology, and the Irish original of the word is anacol ... ‘act of protecting; deliverance; mercy, quarter, safety,’ a verbal noun from the Old Irish verb aingid, ‘protects.’" If that sounds unlikely, try a theory William and Mary Morris turned up. It goes back to a Latin expression used by Roman youngsters who got into trouble: patrue mi patruissime "uncle, my best of uncles." It may be rather more probable that it’s a requirement the person should cry for his uncle in order to be let free. But why uncle?

Interestingly, the earliest examples — found by Dan Norder — are all in the form of a joke. This has a number of forms which appeared in various U.S. newspapers from 1891 through to about 1907 (and which reappeared in the early 1940s), often on the children’s pages. This is the earliest he has found, from the Iowa Citizen of 9 October 1891:

A gentleman was boasting that his parrot would repeat anything he told him. For example, he told him several times before, some friends, to say "Uncle," but the parrot would not repeat it. In anger he seized the bird and half-twisting his neck, said: "Say ‘uncle,’ you beggar!" and threw him into the fowl pen, in which he had ten prize fowls. Shortly afterward, thinking he had killed the parrot, he went to the pen. To his surprise he found nine of the fowls dead on the floor with their necks wrung, and the parrot standing on the tenth twisting his neck and screaming: "Say ‘uncle,’ you beggar! say ‘uncle.’"

Later versions make the reason for choosing uncle as the key word clearer by starting the story "A man whose niece had coaxed him to buy her a parrot succeeded in getting a bird that was warranted a good talker."

The vital question is the same as that regarding the chicken and the egg: which came first, the joke or the children’s call to submit? George H. Goebel, assistant editor of the Dictionary of American Regional English, points out in private correspondence the Iowa Citizen attributes the joke to a periodical called Spare Moments, which was a London weekly of the period. He comments, "An English origin is also suggested by some points of language: ‘a gentleman’ and ‘you beggar’ both sound slightly off to an American ear, and are removed in later newspaper versions of the joke." He concludes, "But as the idiom ‘say uncle’ is apparently strictly American, the joke cannot be an allusion to the idiom, and hence the idiom must be an allusion to the joke."

Few matters are clear-cut in etymology and there’s room for an unexplained transfer of language between U.S. and British English (say somebody taking the U.S. expression across the Atlantic long before it was first written down, which inspired an English comedian to produce the joke, which was then fed back the other way). But the balance of probabilities is heavily weighted towards the American idiom being derived from an English joke.