With my family we have enjoyed the Courtyard Times Square South. If you are on a budget you can walk to a lot of things and enjoy Duke's Deli for inexpensive food as well as a host of reasonably priced places close by. Have Fun. I am sure TJC will have more for you!
If you are in Times Square, Carmine's is also a great restaurant for families. Food is served family style. Huge portions for 4 boys and a tight budget. As far as hotels are concerned, nothing is cheap on points in NYC. A Fairfield in will be 30,000 points and the Marriott Marquis is 40,000 points. I stay in NYC on points a lot and since you are going to burn through many points no matter where you are, you might as well go for the Marquis and get the extra amenities.
In July I stayed at the Courtyard Fifth Avenue, which is on 40th street a few feet from 5th. I was pleasantly surprised by this property. The room was a little tight but very well maintained and decorated. I would recommend this hotel to anyone.
Take your boys to Cooperstown, NY and it is worth the trip. I would recommend the Residence Inn in Greenbush, NY for my stay if visiting Cooperstown. Baseball hall of fame, the drive is beautiful. Staying at the RI, you could also visit the owner (baseball seaon) of the Valley Cats and possibly catch a game. Your sons will enjoy that trip more than anything. I know, I have four boys and to this day (2006 they stayed with me for 2 weeks) they comment on how they loved the trip to Cooperstown.
I recommend staying just across the Hudson in Jersey City at the Courtyard. The PATH trains (subway to Manhattan Ground Zero area) are right outside and you will likely save $100 a night staying here. It's in a relatively nice area, with shopping and restaurants within walking distance. You can be at Ground Zero in 10-15 minutes if you time the PATH train right. Then you can catch a NYC subway to practically anywhere you want to go from there. Also a short walk from your hotel to the river gives you a great view of the NYC skyline.