I've lived in NYC my whole life, and fun varies from kid to kid, but for unique things on a budget, and for such a short trip, I would suggest trying to cluster activities so that you can do a lot in a day without too much back-tracking. For example:
day 1: Metropolitan Museum of Art or American Museum of Natural History and Central Park (one is on Central park West/one 5th Avenue...the Park's east side) then grab a bite in the food court in the downstairs of the Plaza hotel (on the east corner of 59th street and 5th avenue) and visit
FAO Schwartz (toy store in "BIG" ) and maybe hit Dylan's Candy Bar (on
60th st. and 3rd Avenue...owned by Ralph Lauren's daughter)
day 2: Plan to ride the Staten Island Ferry, sitting on the right side of the boat as you board, if you want to glimse the Statue of Liberty. Get the return boat and walk along the water's edge to the South Street Seaport (mostly shopping there, but classic fish market still there, and boats to see.
Ask someone how to get to Chinatown, or just walk under the Brooklyn Bridge going north, and you will end up there. Eat a cheap meal and shop for things you might not have back home....The block one block west of Mott street has cheaper goods then on Mott itself. Buy some hairy red looking fruit from street vendors (rambutan or lychee) they look like things from space, but have a white fleshy fruit that feels like maraschino cherries, and tastes like the love child of coconut and pineapple sort of (don't eat the seed.) Or you can go to the Chinatown ice cream factory anfd have the zen butter or black sesame ice cream or longan sorbet. If not tired, you can meander to Little Italy which is just the other side of Canal street, or further west to Soho.
day 3: Hit anything you always thought you wanted to see in NY.
The subways are really efficient. Ask the attendant in the booth for directions, and ALWAYS ASK ABOUT SERVICE CHANGES! Maintenance often happens on weekends, so trains don't go their usual way sometimes! Keep your cameras concealed, so you don't look like you don't know where you are going, but feel free to ask questions of people who are waiting for trains! Most NYers, the ones who have been here a while, are happy to help if they can! Feel free to PM me with any specific interests. Have a great trip!
ps. At the museums mentioned above, they have a suggested price of admission, but you can actually pay any amount per person...this will make it pocket friendly!