Wow, there are so many places that it really all depends on what you're after food-wise and what constitutes "break-the-bank."
I prefer Peter Lugers when it comes steaks. Hand's down, it's amazing. Cash only there, so be advised.
Then there's Patsy's (legendary, if you can get in), Rao's (legendary), The Palm (legendary), Babbo (Mario Batali's place), Bolo/Mesa Grill (both owned by Bobby Flay), Tribeca Grill (Deniro's place), The View (a circular/rotating restaurant...excellent too), and there's a million other places that are combinations of off-the-beaten-path and hole-in-the-wall that are each outstanding. You'll be in NYC, one of the largest cities in the world, so whatever you want, it's there.
If you get the chance, you'll need to hit Ray's Pizza (the original) and have a Gray's Papaya dog.
Edited to add...
There's also Les Halles (Anthony Bourdain's place), one of which we are lucky enough to have here in Coral Gables. And, very much in the Les Halles style, ther is Balthazar. Long waits without a reservation, but very good.
Alfred Portale's Gotham also comes to mind now. Very, very good.
Also, speaking of Miami-New York restaurants, Joe Allen's in NYC is excellent, and very afforadable price wise. There's one here on Miami Beach as well and I go frequently.