Fouquet's, chez Julien...if you got the money.

Avoid food at Lido and the Moulin Rouge,

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You should consider calling the different restaurants in advance.

I'm not sure if this is done in Paris but a lot of restaurants in cities around France tend to be by reservation only that evening with a fixed special NYE menu for everyone ( including champagne, toasts, etc to really celebrate the new year ) and in many cases they can be cost prohibitive if you are on a tight budget.

You should also take a look at book guides like Frommers ( or you can look online for condensed version http://www.frommers.com/destinations/paris/ ).