The Brooklyn Bridge Park is well worth a trip. Amazing views, a restored old fashioned carousel (Jane's Carousel) and great kid stuff, particularly at Pier 6, which has loads of water activities.
If you walk up to the DUMBO end of the park, there is famous (although I personally think it's a bit overrated) pizza at Grimaldi's and great ice cream in an old lighthouse (Brooklyn Ice Cream Factory) with views across the river.
You can take the NY Water Taxi back to Manhattan - they offer an all day hop on, hop off pass, too. It might be worth checking their times and routes - on the water is a nice way to navigate the city.
You can rent rowboats at the Loeb Boathouse in Central Park, and model sailboats at Conservatory Water (http://www.sailthepark.com)
There's an 18 hole mini golf course in the Hudson River Park (at Pier 25, N Moore Street)
The Children's Museum of Manhattan on West 83rd Street.
The New Victory Theatre specializes in different kinds of theatre (dance, music, acrobats) for kids, all very high quality - you'll need to buy tickets in advance. http://www.newvictory.org
Outdoor kids movies in Hudson River Park
For some inexplicable reason, my kids used to think riding the Roosevelt Island Tram was really fun (a good activity if you're planning to go to Serendipity 3 (East 60th Street) for frozen hot chocolate as it's nearby.
And if all else fails, or the weather is awful, there's always bowling and ice skating at Chelsea Piers (bring your credit card)
ETA: some portions of the NY Aquarium at Coney Island are still closed, but if you have the time, it's a still worth a trip. And riding the D train out to Coney Island is an... interesting experience.