Santa Claus as a Symbol of Modern Christmas:
Christians treat Christmas as a Christian holiday, and it certainly started out that way, but we can tell a lot about the real nature of holidays by how they are represented in popular culture. The most common, popular, and recognized symbol for Christmas today isn't an infant Jesus or even a manger scene, but Santa Claus. It's Santa who graces all the ads and decorations, not Jesus. Santa Claus is not, however, a religious figure or symbol — Santa is an amalgam of a little bit of Christianity, a little bit of pre-Christian paganism, and a whole lot of modern, secular myth-making.
Santa Claus, Christian Saint?:
Most assume that the Santa Claus of modern Christmas is based on a Saint Nicholas in Christianity, but any connection is tenuous at best. There was Nicholas who was bishop of Myra in the early 4th century and who stood up to anti-Christian persecution, but there's no evidence that he died for refusing to renounce his faith. Legend has it that he did good works with his family's fortune and he became a much-loved figure in most European cultures. Over time, he was given attributes of pagan figures who were popular during winter festivals.
Washington Irving and the Invention of Saint Nick:
It is argued by some that the modern Santa Claus was basically invented by Washington Irving who, in a satirical history of New York, described alleged Dutch beliefs about Sinter Claes, or Saint Nicholas. Most readers accepted Irving's descriptions as fact and helped people to eventually adopt many of the beliefs and traditions attributed to the Dutch, though not during Irving's lifetime.
Clement Moore and Saint Nicholas:
Most contemporary ideas about what Santa Claus does and looks like are based on the poem The Night Before Christmas by Clement Moore. That has two things wrong: it's original title was A Visit from Saint Nicholas, and it's unlikely the Moore really wrote it. Moore claimed authorship in 1844, but it first appeared anonymously in 1823; explanations for how and why this happened are implausible. Some of this poem borrows from Washington Irving, some parallels Nordic and Germanic myths, and some may be original. This Santa Claus is completely secular: there's not a single religious reference or symbol to be found.
Thomas Nast and the Popular Image of Santa Claus:
The poem attributed to Moore may be the basis for current conceptions of Santa Claus, but Thomas Nast's drawings of Santa Claus during the latter half of the 19th century are what engraved a standard image of Santa Claus into everyone's mind. Nast also added to the Santa character by having him read children's letters, monitor children's behavior, and record children's names in books of Good and Bad behavior. Nast also seems to be the person who located Santa Claus and a workshop for toys a the North Pole. Although Santa here is smaller, like an elf, the image of Santa is basically fixed at this point.
Francis Church, Virginia, and Santa Claus as an Object of Faith:
In addition to Santa's visual appearance, his character also had to be created. The most important source for this may be Francis Church and his infamous response to a letter from a little girl named Virginia who wondered if Santa really exists. Church said that Santa exists, but as everything but a real person. Church is the source of the idea that Santa is somehow the "spirit" of Christmas, such that not believing in Santa is the same as not believing in love and generosity. Not believing in Santa is treated like kicking puppies for fun.
What's so Christian about Santa Claus?:
There's little to nothing about Santa Claus that is either uniquely Christian or broadly religious. There are certainly a few religious elements to Santa, but he can't be treated as a specifically religious figure. Almost everything that people today understand as part of the Santa Claus myth was invested in this figure fairly recently and, it appears, for entirely secular reasons. No one took a beloved religious icon and secularized it; Santa Claus as a Christmas figure has always been relatively secular, and this has only intensified over time.
Because Santa is the central figure for Christmas in modern America, his basically secular nature says something important about Christmas itself. How can Christmas be essentially Christian when the leading symbol of Christmas is essentially secular? The answer is that it can't — while Christmas may be a religious holy day for many observant Christians, the Christmas holiday in the broader American culture isn't religious at all. Christmas in American culture is as secular as Santa Claus: it has some Christian elements and some pre-Christian pagan elements, but most of what makes up Christmas today was created recently and is basically secular.
The question of "what's so Christian about Santa Claus?" is a stand-in for the larger question of "what's so Christian about Christmas in modern America?" The answer to the first helps us answer the second, and it's not an answer which many Christians will be pleased with. Not liking the situation won't change anything, though, so what can Christians do? The obvious route to take is to replace secular observances of Christmas with religious ones.
So long as Christians continue to focus on Santa Claus coming to town to deliver gifts rather than on the birth of their savior, they will remain part of what they see as the problem. Dispensing with, or even just limiting, the role of Santa Claus and other secular elements of Christmas probably won't be easy, but that only demonstrates just how deeply enmeshed in secular culture Christians have become. It also reveals just how much of their own religious Christmas they have abandoned in favor of secular celebrations. In effect, the harder it is the more this shows that they need to do it if they want to claim that Christmas is religious rather than secular.
In the meantime, the rest of us can enjoy Christmas as a secular holiday if we want.
See Tom Flynn's The Trouble with Christmas for more on this.