If by incite you mean an incentive for action, then it would be the fact that Della had been saving and saving for months, and all it amounted to was $1.87, clearly not enough to buy Jim a nice present. At this realization, Della decided that "There was clearly nothing to do but flop down on the shabby little couch and howl." But after this good cry, a light bulb goes off and she "pulled down her hair and let it fall to its full length." She would sell her hair, hair "in which they both took a mighty pride." She does, buys him a chain for his watch, only to discover that Jim sold his watch to buy her a set of hair combs.
What started this chain reaction though was the description of how Della realized she would not have enough money to buy Jim a present that she felt reflected her love for him-this prompted her to action, and incited the rest of the events.
The inciting incident of the plot of "The Gift of the Magi" occurs when Della starts crying after counting her paltry $1.87, which is all she has for a Christmas present for her husband, Jim.
In the plot of a short story, the inciting incident is that first event which indicates that the narrative's problem has begun. Thus, when Della counts her change and finds that she only has $1.87 after having saved for months, the story commences. In despair she begins to cry because she has insufficient funds for a Christmas gift for Jim, her beloved husband, and Christmas will soon arrive. But, Della finishes her cry, and after looking in the pier glass and wiping away her tears, she reaches a decision. Della puts on her coat and old brown hat and heads toward Madame Sofronie's where she sells her luxurious hair. Then she buys a Christmas present for Jim.
The ironic twist in O. Henry's narrative is that Jim Dillingham has sold his watch for a set of lovely combs for his wife's hair while she has sold her beautiful hair for a chain for Jim's watch. All this action follows the inciting incident of Della's discovery that she does not have enough money with which to purchase Jim a Christmas gift.