The museum of Natural History is nice & probably the most kid-friendly museum in manhattan. There's also the met (classic art), and the moma (modern art), both on Central Park East, I believe. If you want to get out of just manhattan, in Brooklyn you can go to the aquarium (which really isn't that impressive.. it's smaller than I had expected). If you get tired of walking the circle line bus tours are nice. If you like just wandering around looking at people/places, check out City Hall, walk across the Brooklyn Bridge, check out China town (Canal St.).
Honestly, when I have friends come to the city, the highlight of the weekend is generally the food/nightlife. There are some amazing restaurants that you just WON'T find anywhere else. Whatever you do, don't go to a chain. Some of the neatest restaurants are the little holes in the wall. However, there are some really nice fancier places that are very family friendly (lax dress code, noisy, but amazing food and a hefty price). If they have internet access,
www.menupages.com is indispensable. The zagat's restaurant guide is also so helpful (You can just look at one in Barnes & Nobles without actually buying it.. I did that for a while when I wanted a nice restaurant
)
I'd probably avoid the subway in the summer, especially with a kid. The 4/5/6 line is super crowded, especially.. if you're on the east side. But regardless of the line, it gets really hot in the stations (it's not so bad when the humidity is low.. but if its 80% or so, it's killer).
If it's this weekend, I might avoid some of the more touristy stuff, like going up in the Empire State Building and stuff like that. People will be flocking in. Also, more people will probably be out this weekend than usual, so be warned for food.
Another thing to check out could be the TIX booth in Times Square (there's a similar thing down at the South St. Seaport) where you can get day-of discount tickets for Broadway shows (although on a weekend, they'll only have the less-popular ones or off/off-off broadway shows).
Also, for one of the best views of the city, take the Staten Island Ferry (down near battery park) over to Brooklyn. It's free and pretty relaxing. Seriously, it's great.