One of the first times I ever ate brunch was a New Year's Day - it was at the Chase Park Plaza, in what was then, I believe, the Hunt Room. I was amazed that there were enough people interested in brunch after the annual late-nighter that the hotel had gone for it. (I was young - while I wasn't a heavy partier, I assumed pretty much the rest of the world was.)
Atomic Cowboy is doing brunch New Year's Day, and every Sunday. I wrote about it as part of the Tour de Toast series at St. Louis Magazine's Dining blog. I'm not sure how quiet it will be this New Year's Day. But there's plenty of the sort of food that helps metabolize the leftovers from too much ethanol having been imbibed.