"Auld Lang Syne" is pretty good, but New Year's Eve could use another carol or two, given that it's kind of a sequel to the most carol-filled holiday of all time.
Arthur Rankin Jr. and Jules Bass had a hit on their hands with their stop-motion animation special Rudolph the Red-Nosed Reindeer, so some 12 years later, they came up with Rudolph's Shiny New Year, which had the obvious choice of holiday for a follow-up. Instead of a Burl Ives snowman, it featured Red Skelton as Father Time, and Rudolph had to hunt for the big-eared Happy New Year baby that was kidnapped by a jerk vulture. With the help of Santa Claus, Ben Franklin, a caveman, and a Scottish knight, they save us all from having to live December 31 forever.
It didn't quite have the staying power of the original Rudolph story, but since New Year's Eve has a shortage of songs, here's one from the end of the story that might do in a pinch.
"Have a happy, have a happy, have a hap-hap-happy new year!"
Andy Hunsaker has a head full of sitcom gags and nerd-genre lore, and can be followed @AndyHunsaker if you're into that sort of thing.