The gift Jim and Della give to each other is the gift of sacrifice and unconditional love. Della sacrifices her long hair and Jim his antique pocket watch, which are the prized possessions for each of them. Ironically, of course, Della's sacrifice renders Jim's gift of tortoise-shell combs useless; Jim's sacrifice of his watch makes Della's gift of a watch chain useless as well.
The point of the story--the lesson that transcends the story and even time--is that the best gifts are given from a pure heart. No matter the cost, if a gift is given in sincerity and love, it is a perfect gift. The title of the story is a Biblical allusion to the wise men who brought gifts for the Christ-child. The gifts, which included frankencinse, gold, and myrrh, were traditional gifts presented to a king. Since the word gift in the title of Henry's story is singular, it can not refer to both Jim's and Della's gifts, but instead to the gift of unselfish love they present to one another.
They give each other gifts for Christmas that can be described (watch chain, hair combs), but the real gift that they gave each other was sacrificing their own most prized possession to pay for the gift they gave to the other person. Therefore, the real gift was that they each loved the other so much that they were willing to sacrifice for them. It wasn't about the monetary value of the gifts, but the depth of love they felt for each other. The Magi are the wise men in the Bible who brought gifts to the Christ child. They traveled a long distance from the east to show honor to Jesus and bring him symbolic gifts honoring his unique position. The theme of the story could be that it's not the monetary value of a gift that matters, but the true motivation and level of sacrifice that gives the gift value.