Courtesy of fishbulb42, a recent discovery: Wonderful World of Joey - What Sweet Child o' Mine Is This?
Hugh Blumenfeld - Long-Haired Radical Socialist Jew
Rev. Run & The Christmas All Stars (feat. Mase, Puff Daddy, Snoop Doggy Dogg, Salt 'n' Pepa, Onyx & Keith Murray) - Santa Baby
More seriously, I've always been partial to The Nutcracker (the suite, or the full ballet). And I've loved the Carol of the Bells ever since I first saw the Claymation Christmas Special. But those technically aren't songs.
We Three Kings (Chris Isaac does a great version)
White Christmas (I saw a great stage adaptation of the movie last year)
I have a great album of traditional Finnish Christmas songs sung by Sari and Mari Kaasinen (of Värttinä).