When the UN was in session and rates were sky high in Manhattan, I stayed at the Courtyard Jersey City Newport recently. Rooms are good size for a CY. Not sure if rollaway beds are permitted, you would have to check with the hotel. It is literally adjacent to the PATH station with very easy access to NYC. It is a CY, so very slim odds of a suite upgrade and no lounge. There is also a Westin in the same area though I haven't stayed at it yet. If the Manhattan points rates are very high, Jersey City tends to be a bit lower. Book early or it can get just as expensive as Manhattan. Finally, Jersey City area has some interesting ethnic restaurants (may not be of interest with very young kids). Hoboken is another option with a PATH stop thought I haven't stayed there yet.
Count me as a dissenting vote on not staying in downtown Manhattan. You have the 911 Memorial, Eataly (which is touristy but fun), The South Street Seaport, Staten Island Ferry, and Statue of Liberty Ferry all in the area. The Manhattan Chinatown, Brooklyn Bridge, and the Tenement Museum are also close by. Finally, you can get anywhere in the city very easily by subway. Depends on your comfort level on using the subway with your family. Personally I am fine with it other than late at night. For late at night, Uber's should be less expensive since there is less traffic. The benefit of staying downtown is you can get properties with larger rooms at lower price points. New York Marriott Downtown and the Residence Inn at 4 Maiden Lane (not the newer one in the Financial District) have good size rooms (at least for NYC) at relatively reasonable price points.
If you want luxury properties with good lounges, your options will be much more limited and higher price points. Nothing wrong with that, just a personal decision on cost/benefit.
Finally, Times Square is a crowded zoo virtually 24x7. I do not recommend staying there with young children (well really at all).
Good luck and have fun!
--Jon