Ethos, pathos and logos are persuasive appeals. Pathos and logos are used in the story. Even in a short story, an author has a message to send (the theme). O’Henry is trying to convince us to be careful with our choices, and that love is the most important gift.
Ethos is the idea that you believe a person because the person is an expert. In a short story, this one is not as applicable. The closest thing we have to ethos is O’Henry’s use of the Magi, famous gift-givers. The Magi are not telling the story though.
Pathos can best be interpreted as emotion. I think of it as sympathy. This is a very sappy story. We feel for the struggling young couple, and they are presented very sympathetically. We are convinced that they could have made better choices, but also that they love each other and will be all right because of that.
Logos is logic. This means you use reason to make your argument. This is actually a very ironic story. Delia sells her hair to buy Jim a chain for his watch, and Jim sells his watch to buy her combs for her hair. Logically, each one sold something to buy something that the other could no longer use. We are struck by the logic, it makes us think, and hopefully we are convinced!