The allusion to the Magi of the Bible, extends to the idea that these men traveled from afar to be in the presence of the greatest gift of God to mankind, His only Son:

For God so loved the world that He gave his only begotten Son, that whosoever believeth in Him, shall not perish, but have everlasting life. [John 3:16]

Clearly, O. Henry's story is one with a very Christian theme, the theme of love as meaning that one is willing to sacrifice one's own desires for the happiness (or safety) of another.  Like the Magi, Della and Jim understand the true meaning of love, and they understand that sacrifices are often necessary in order to experience true love.  Like the Magi who traveled from great distances under strenuous conditions, Della sacrifices her prized possession of her luxurious hair in order to buy a present for Jim, and Jim does likewise for Della; in the end, they give each other the richest gift of love, just as the Magi bring the baby Jesus rich gifts.

The magi are significant for two reasons.  One is that the use of the term shows that Jim and Della are truly wise.  The other is that using the term is a bit ironic.

The term is ironic because the wise men in the Bible were these really rich people who could afford really expensive and wonderful gifts.  By contrast, Jim and Della are quite poor.

The magi are also significant because using their "name" enables the narrator to talk about how Jim and Della are truly wise because of the way they show their love.

What does the author mean by the Magi in the story "The Gift of the Magi"?

In O. Henry's story "The Gift of the Magi," the word "magi" has two different meanings. On the surface level, it refers to the "wise men" spoken of in the Gospel of Matthew who came from the East bearing gifts for the infant Jesus (Matthew 2:11). This is a central part of the Christmas story; most people know about the "three kings" and assume that the reason we give gifts at Christmas is because these worshippers brought gifts for the Christ child at the first Christmas.

O. Henry plays off this Christmas tradition in his Christmas story. In his story he tells of Della and Jim, a young newlywed couple who are very much in love. Unfortunately, they don't have enough money to buy Christmas gifts for each other. Della ends up cutting off and selling her long hair to buy a chain for Jim's gold watch, which he has pawned to buy combs for Della's long hair. Henry points out how "foolish" the young lovers are for doing such a thing, but he ends by saying, "Of all who give gifts, these two were the most wise. ... They are the magi." By this he makes a point by way of metaphor that people who give from a heart of self-sacrificing love are the wisest givers--just like the magi of old who traveled a long journey to worship Christ. So Della and Jim, although the narrator initially brands them as fools, are in reality magi--they are wise enough to forget themselves and give from their hearts. 

What is the role of the Magi in O. Henry's story "The Gift of the Magi"?

The Magi are not literal figures within O. Henry's "The Gift of the Magi." Rather, the title is the reference to the presentation of gifts by the three wise men (or magi) to the infant Jesus.

In this story, a young married couple makes great personal sacrifices to buy each other beautiful presents for Christmas. Della Young has only $1.87 to buy her husband, Jim Young, a gift. In order to purchase a platinum fob chain for Jim's gold watch, she sells her hair to Madame Sofronie for $20.00. Meanwhile, Jim sells that same gold watch so that he may afford a set of pure tortoise shell combs for Della's hair.

Both have received gifts which are now ultimately useless to them, but the sacrifices that made those material items useless are in all actuality the more beautiful gifts... those acts of selflessness are testimony to the love that this couple has for each other.

The story concludes with a discussion of the parallels between these two:

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 the wisest. They are the magi.

Thus, it is illuminated that Della and Jim (and all others who sacrifice out of love) are the magi: the greatest of givers and the owners of the supreme knowledge of devotion. 

Who are the Magi? Why does O. Henry compare his story "The Gift of the Magi" to them?

The Magi were the wise kings who visit newborn Jesus and bestow upon him gifts of Frankincense, myrrh, and gold.  According to the Book of Matthew in the Bible, the magi follow the star from the East to Jerusalem, looking for the one born to be the king of the Jews.  When King Herod hears of their arrival, he wants them to find out more about the prophecied Christ.  When the magi finally reach their destination in Bethlehem, they see Jesus with Mary and worship him, bestowing their gifts to the newborn king.  The magi, full of love for Jesus, do not return to report back to Herod, but rather go home a different way.

"The Gift of the Magi" as a title suggests story about a gift given from the heart, free of unselfish desires or motives.  Like the magi in the Bible, Delia and Jim Young both offer gifts to each other in complete sincerity and out of love.