O.Henry is referring to the act of selfless giving when he speaks of The Gift of the Magi. In this story, we see a fairly poor couple sacrifice their greatest possessions for the betterment of each other. Della cuts her long, flowing hair, and Jim sells his family heirloom watch. They then give one another gifts that require what they sold in order to work: combs for Della's hair, and a chain for Jim's watch. This type of selfless giving is the "gift" that O. Henry is talking about in the story's title.
This is the story of a young couple who isn't able to afford Christmas gifts for one another. In O'Henry's The Gift of the Magi, he writes:
"The magi, as you know, were wise men--wonderfully wise men--who brought gifts to the Babe in the manger. They invented the art of giving Christmas presents. Being wise, their gifts were no doubt wise ones, possibly bearing the privilege of exchange in case of duplication. And here I have lamely related to you the uneventful chronicle of two foolish children in a flat who most unwisely sacrificed for each other the greatest treasures of their house. But in a last word to the wise of these days let it be said that of all who give gifts these two were the wisest. O all who give and receive gifts, such as they are wisest. Everywhere they are wisest. They are the magi. "
Here he tells you that the "most unwisely sacrificed for each other the greatest treasures of their house". What he means is that the greatest gift you can give is that of yourself, in this story the young couple sacrificed their greatest treasure to be able to give each other a Christmas gift.
He isn't referring to anything material; he is referring to self-sacrifice for love.