Why do bullies insist their victims "say uncle"?

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Why: In West Side Story, Tony twists Riff's arm until he says "I hate living with my buggin' uncle! Uncle!! UNCLE!!"

Answer: It is strictly an American phrase, though it may have originated somewhere else. One possibility:

• It comes from 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."
  

Another theory is that during the Roman Empire, children beset by a bully were forced to say, "Patrue, mi Patruissimo," or "Uncle, my best Uncle." Some speculation regarding just why bullies wanted their victims to say that include:

• It was a way of making the victim call for the help of a grownup, thus proving his helplessness.
• It forced the victim grant the bully a title of respect. After all, in Roman times, your father's brother had nearly the same power and status as your father. The form of "uncle" (patrue) used here refers specifically to a paternal uncle. A mother's brother (avunculus) occupied a lower rung in the patrilineal society.
  

Source: WorldWideWords, Phrases.org

The More You Know: Here is a "joke" from an 1891 edition of Iowa Citizen:

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.’”