"Christmas Wrapping," The Waitresses. You still hear this one a lot, for good reason. It's probably the best song this short-lived band ever came up with (the only other contenders being "I Know What Boys Like" and "Square Pegs"), and it's a fun one to put on at a Christmas party.
"It's A Big Country," Davitt Sigerson. Not only did this guy sing, but he also used to produce (an album for David Johansen in the early 1980s that I have), and write reviews (for the Village Voice, and perhaps also for Rolling Stone). I have no idea what happened to him, but I haven't seen his name in years. This song is a bit sappy, but I always liked it - essentially, a musical Christmas card to members of his family, strung out across the country.
"Christmas on Riverside Drive," August Darnell.
"Christmas Time in the Motor City," Was (Not Was).
And of course, the legendary "Christmas With Satan," by James White. Not exactly your run-of-the-mill holiday fare, but certainly memorable.