I did two trips to NYC with my mom at ages 10 and 12. Things I remember:
carriage ride
FAO Schwartz
Tiffany's
Steuben
Broadway shows
Statue of Liberty
a private museum in a mansion
Where we ate figured prominently: Russian Tea Room, Tavern on the Green, a restaurant at the base of the Brooklyn Bridge that I can't remember the name of, Plaza for tea, ice cream at Serendipity.
If there's still ice, go ice skating either in Central Park or Rockefeller Plaza.
It might be fun to watch some quintessential NYC movies before going. Big. Breakfast at Tiffany's. Ghostbusters. Night at the Museum. Cruel Intentions, if not too mature.
Some lists:
http://onthesetofnewyork.com/home.html
http://en.wikipedia.org/wiki/List_of_films_set_in_New_York_City