Walking Distance 61st & Madison
Resturants, take out, deli.
Will be in NYC over a weekend with my kids(7 & 10).
Mangia is on 57th between 5th and 6th. It's mostly takeout with some tables, though there's a full-service restaurant upstairs. Sandwiches, salad bar, that sort of thing with an Italian spin, but a much better quality than the typical deli.
Viand Coffe shop on Madison and 62nd - quintessential NY diner experience with excellent burgers, fresh turkey, etc.
Le Pain Quotidien - Lex between 63rd and 64th. Check below for the link to the menu - very informal with communal seating.
Menchanko Tei - 55th betw 5th and 6th..Japaneese Noodle shop for great soup.
Im not accustomed to selecting places for kids so hopefully these will suffice.
When my niece & nephews (chowhounds all) were around that age, I took them to the cafe at Arizona 206 (60th between 3rd & 2nd). They loved it.