Can you provide exact dates for check-in and check-out?
How many rooms would you require?
Most hotels in New York City limit occupants to two adults. Children can be accommodated in existing bedding, but no cots, etc. can be added. The rooms are small and there just isn't room for more bedding. The cheapest thing I found in New York City was $150 per night for one room.
If you can go outside the city (CT, NJ) the rates are somewhat lower. Would you consider a B & B? I found a couple for about $85-95 per night for two people, but they aren't downtown.
If you're willing to stay in Secausus, NJ (4 miles from NY City center), here is a possibility where you'd only need one room (La Quinta Inn and Suites):
It's actually closer to the city center than staying in any of the boroughs.
Here's one in Bergen, NJ,(Holiday Inn Express) just 3.1 miles to Times Square:
Keep in mind that hotel taxes in the New York area are nearly 15%, and they will add on about $10 per day for a cot for the child.
When I'm looking for a hotel, I always try trivago.com first They will find the cheapest alternative that matches your needs. Do always pick a hotel that offers free breakfast. It will save you at least $10 per person per day.
As for what to do, don't miss the Statue of Liberty and Ellis Island, a walk around Times Square, Central Park, Bronx Zoo, the Chrysler Building if you're interested in architecture (it's beautiful), Empire State Building, the Museum of Modern Art. For lunch, don't miss a hot dog at any of the street vendors. Delicious, and it's the cheapest meal in town. Plan to have your biggest meal in the middle of the day. Most restaurants serve the same food as they have at dinner, but the prices are considerably lower.
I hope this helps some. Have fun on your trip.