There are actually quite a few places shown as open on New Year's Day on Opentable. Some of the better mid-priced restaurants open include Aria, Bistro Margot, Blackbird, Cafe Spiaggia, the various Mia Francesca locations, Le Colonial, Mercat a la Planxa, and Real Tenochtitlan, and many of the steakhouses and seafood houses (Capital Grille, David Burke's, Joe's, McCormick & Schmick, Pete Miller's, Rosebud, Ruth's Chris, Smith & Wollensky, Sullivan's, Wildfire).
As for Chloe's idea (which is also a good one), you can find the Metromix listings for New Year's Day by clicking
here.