For an old school diner, I like Lexington Candy Shop when I worked nearby, although it’s been a while since I was last there.
@KimCardiffian this one is relatively close to the Metropolitan Museum of Art and Guggenheim.
Three Guys at 960 Madison is another diner that’s not too far from the Museums (closer to the Frick & Neue Galerie than the Met) http://places.singleplatform.com/3-guys/menu?ref=google
Viand also has a few locations. It’s got a good greasy spoon type breakfast. I usually get omelettes. Not worth going too far out of your way, but might be handy, and the atmosphere is old NYC. I usually visit the location at 673 Madison.
Some diners are cash only. Viand was cash only last time I was there.