I'll agree with Juniors, or the Edison Cafe, located in the Edison Hotel. Nothing fancy. It actually has an entry from the street, but you may come across a broadway performer or two eating there.
Broadway Shows I've Seen: Hairspray, Chicago, Little Shop of Horrors (2003), The Wedding Singer, Spamalot, Riverdance, Rent, Beauty and the Beast, Spring Awakening, Wicked, Legally Blonde, Phantom of the Opera, Sweet Charity (revival), Drowsy Chaperone, The Lion King, Dreamgirls(2010 Tour).