A few of our favourite things, with the kids when we lived there:
Childrens Museum of Manhattan, round the corner from the Natural History Museum on the Upper West Side
Dinosaur section of the Natural History Museum
Planetarium section of the Natural History Museum
Central Park Zoo (make sure you visit the smaller section too, across the footpath from the main Zoo, which is very good for smaller children)
Walk up to the small boating lake and hire out a radio controlled boat to sail - or watch others
Alice in Wonderland statue at the boating lake
Staten Island Ferry for great views of Statue of Liberty (about $1)
Get Time Out for the Weeks listings (has Kids section)
Hire a rowing Boat out on the Boating Lake - great views of Dakota building where John Lennon lived
Eat at EJ's diner (Upper East and West Side)
Catch an off-Broadway childrens show
Star spot!