A variety of recommendations, to suit various prices & tastes.
If money's no object, go to Le Cinque at the Hotel Georges V. It's brilliant: the room is stunning, the service is perfect, and the food is inventive and delicious.

I like to eat at ambitious little neighbourhood places off the beaten tourist track. My current faves are Le Villaret (13, rue Ternaux, 11e) and the rather more rustic Le C'Amelot (50, rue Amelot, 11e).

If you want cheap, touristy Parisian fun (but the best kind), go to Cremerie Polidor (41, rue Monsieur-le-Prince, 6e). New Years Eve--more properly, St-Sylvestre--is a big deal there, and the back wall is decorated with huge jeroboams and methuselahs (or something--I'm lost above a magnum) of champagne finished off on previous St-Sylvestres.

I'm sure any of the major bistros & brasseries would be a hoot on St-Sylvestre. I like Bofinger and Balzar, personally.