If you find yourself in TriBeca, go to Bubby's, a restaurant that manages to be incredibly hip, yummy & child-friendly all at the same time. It's on Hudson & North Moore. They've a great kids' menu offering variations on the "southern home style" cooking that grown-ups eat, a dessert menu that contains about 20 different famously good pies. And kids get a placemat with mazes & word-gamey activites, with crayons, to boot. They can leave with balloons, too, if they like.
A famous old-school child friendly restaurant is the formerly Truman Capote-inhabited "Serendipity 3" on 60th street between 2nd & 3rd Aves. It has a total OTT Art Nouveau look, with Tiffany lamps dangling from every corner, and an incredibly eccentric menu that appeals to adults & kids alike, including foot-long hotdogs and jumbo frozen hot chocolates & cascading frozen pink lemonades. It's almost impossible to get a booking, though, so arrive as close to opening as possible (11am) and go there for lunch one day. Dinnertime is just too much of a zoo IMO.
I'm a transplanted New Yorker, by the way, and I happen to be living here temporarily once again. . .so I face that question daily: what to do with the kids? I've concluded that London is a more child-friendly city than NYC: it's easier to haul kids around in London, for one thing, and activities for kids are unnecessarily expensive here. A entry ticket for me & my 2 kids at the Museum of Natural History (recommended!) cost $60, which is about 30 pounds. Let's face it: only people with plenty of money can take their kids there--and that's just unacceptable. And there's a drop-in centre near where I live which is a great place for the under 5's. . .but the entry fee for an adult and 2 kids is well over $20. Compare that to the pound or whatever you have to pay to attend a (better!) drop-in centre in London. I mention all this just so you're prepared!
One activity that hasn't been mentioned yet is the Merry-go-Round in Central Park. . .it's a little hard to find, but it's the real deal: a proper, old-fashioned machine. A ride is not terribly expensive, so I take the kids on it again and again and again. If you're going in winter, though, it won't be open.
I also recommend the Metropolitan Museum. Ask for the map for kids. . .it contains the top 10 or so exhibits that kids love, including an out of the way room that replicates a Japanese garden, replete with pond and swimming fish! The Temple of Dendor there is a big hit, too.