Well, obviously you have to watch the weather, but the Central Park Zoo is fantastic. Central Park is an expensive area to stay in, though, so be sure you have a good reason for staying there or that you have an excellent hotel deal. Definitely use the on line services that are up-to-date on last minute openings - Trip Advisor, Hotwire, and also Groupon, etc. Not sure how long you will be there, but don't be afraid to stay in Brooklyn or Queens, and take the subway across.
Do the Empire State Building. Skip the Statue of Liberty in the dead of winter - it's a historic landmark but it's a tough water crossing to a wind-swept island, best in the spring or summer. Take the subway from your hotel to any of your destinations, doesn't matter where, just for the experience. And take a NYC cab if he has never done it.
What does he like? The Museum of Natural History and the Hayden Planetarium are unbelievable. Not sure if you think he is old enough to watch "Night at the Museum" (PG) or programs featuring Neil Degrasse Tyson (director of Planetarium) but those experiences can make the "real thing" very meaningful for a kid. The Intrepid is an aircraft carrier on the Hudson that has a museum connected to it.
If it's incredibly mild, by some miracle, consider walking across the Brooklyn Bridge. If he likes theater, look for an appropriate Broadway show - there are 2-for-1 tickets on the day of the show.
Chinatown is more diverse than it once was - Thai, Korean,Vietnamese, etc. The main streets to look into are Mott, Broome and Canal. There are plenty of shops for tourists and a zillion restaurants.
Little Italy has more festivals and events in the really good weather, I think. But do contact the NYC tourism office and see what they can offer you, given the length of your stay, the age and interests of your child, and your budget.
South Street Seaport can be fun too but check on ships during the winter. Lots of food options on the Lower East Side, including classic Jewish delis and markets.
Don't forget Harlem - lots of character and history. Same with the outer boroughs. Just keep your trip manageable - you don't want it to be a whole lot of hustling from one area to the next - choose a few things and enjoy them thoroughly.
Are you a AAA member? They have a free travel service and all kinds of coupons, flyers, and guidebooks with updated hotel and restaurant ratings.