Wordorigins.org

To say or cry uncle is to call for mercy, to acknowledge that one is defeated. It’s American playground slang dating back to the early 20th century. From the Chicago Herald-Examiner of 1 October 1918:

Sic him Jenny Jinx—make him say ‘Uncle’.

The phrase may come from the punchline of a joke that was popular in the 1890s. Douglas Wilson (one of this site’s regulars) discovered this version of the joke in the Iowa Citizen, 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.’[sic]”.

Say uncle is sometimes claimed to come from the Irish anacol, meaning mercy or quarter, the lack of currency in Britain explained by the term being brought to America by Irish immigrants. Despite the similarity in sound and meaning, there is no strong evidence to support this conjecture.

(Sources: Oxford English Dictionary, 2nd Edition; ADS-L; Newspaperarchive.com)