The most obvious use of hyperbole in "The Gift of the Magi" occurs when the narrator describes Della's and Jim's evaluations of their two treasures—her long, luxuriant hair and his gold watch.
Had the Queen of Sheba lived in the flat across the airshaft, Della would have let her hair hang out the window some day to dry just to depreciate Her Majesty's jewels and gifts. Had King Solomon been the janitor, with all his treasures piled up in the basement, Jim would have pulled out his watch every time he passed, just to see him pluck at his beard from envy.
This is the wildest kind of hyperbole. This passage claims Della's hair is more precious than all the Queen of Sheba's jewels and her fabulous gifts to King Solomon. The excerpt also says Jim's gold watch is more precious than all the treasures of King Solomon.
O. Henry, of course, does not intend his hyperbole to be taken at face value. He only wants to convey an idea of how much Della treasures her beautiful hair and how much Jim treasures the fine watch he has inherited from his father, who inherited the watch from Jim's grandfather. O. Henry wants the reader to appreciate the sacrifices the two young people make when they part with their only treasures. The author also introduces the biblical motif with the Queen of Sheba and King Solomon. Their famous meeting is described in the Old Testament in 1 Kings 10. The contrast between the Old Testament monarchs and Della and Jim serves as another way of illustrating their poverty.