Wasn’t it one of Santas other reindeer? I’m guessing Donner.
I would guess Blitzen, so named for his propensity to party like a maniac. This was never publicly known, but Blitzen battled the bottle for quite some time. In a drunken state, he had a one night stand with Rhoda the Reindeer, Rudolph’s mother. The famous red nose was a result of Blitzen’s constant state of inebriation, leaving poor Rudy with a chronic case of rosacea.
Donner.
And i bet you don’t know his mother’s name.
It was Mrs. Donner.
Aww, and here I thought Santa got really drunk one night and … well … Santa does have a red nose :D. I’ll shut up now.
Actually it was Adolf the Brown Nosed Reindeer.
He didn’t make the first team though, because while he could go as fast as the others, he couldn’t stop as fast.
Zoggie is right. According to the Rankin/Bass special, Rudolph’s father was Donner. Which is weird 'cause I thought name was “Donder”.
Blech.
That should read:
“…thought his name was “Donder”.”
Need coffee.
T’was Elvis.
**E:**Thang you! Thang you verra much! Ms. Reindeer. Gotta go Viva Las Vegas, now, ya’ll heah?
Ms. Reindeer: OH! EEELLLLVVVVIIIISSSS! (pant gasp)
Named after the party, I assume. Just don’t eat the burgers. 
Trion,
Originally there was a pair of reindeer called Dunder and Blixem. Now they are mostly known as Donner and Blitzen. Snopes has a good page about how this happened.
Cool. Thanks Amok.
Seeing as how reindeer (or caribou, same species) are the only members of the deer family in which both sexes get antlers, how do we know all of Santa’s reindeer are male? Maybe one of the eight is Rudolph’s mother, too. I vote for Vixen.
[insert emoticon of extreme sympathetic look]
Kids still out of school, huh?
According to a book that came out last summer, called Rudy!, he was “a stickup artist who robbed a milkman at gunpoint, then took over as muscle for a relative’s loan-sharking business once he was sprung from prison.”
Rudolph’s father? Some ad exec at Montgomery Wards 
Well, guys, I have to come out and say it.
You see, years ago, when I was stationed in the Arctic Circle, there’s just not a hell of a lot to do up there except drink. Well, one night I was tying one on at The Elves’ Hole where I came across what I thought was the most beautiful sheep I had ever laid eyes on.
You know, it’s the same old story…we went back to my place, and in the morning, I was shocked to find a reindeer in my bed.
Three weeks later, I was back in the same bar, when Vixen (you were right, yabob) walked in, a bit…well, knocked up.
Fortunately, I got a transfer two days later, and I never saw her again.
Wow. Rudolph is one lucky little reindeer. Everyone is taking credit for his paternal…ship.
Very different from…“No way, the lil maroon colored mutant ain’t mine!”
Vixen’s very cool. Remember in The Year Without a Santa Claus? She could lift two elves and fly. And rescue them from Heat Miser! I wouldn’t be suprised to learn that Vixen really gave birth, then gave up the newborn Rudolph to her sister, Mrs. Donner, so that she could continue flying with the team. Yup, it happened. (They couldn’t show that on the special though, because…well, the kids, they wouldn’t understand.)