Times Square will be a zoo during that week (it is always a zoo). Where you should stay depends a lot on what you want to see. The subway is a good way to get around easily in NY at a reasonable price. If you're willing to do that, there is a Holiday Inn in the Financial District which would be convenient for visiting The Statue of Liberty. It also has good subway access. There are also Holiday Inn's in Long Island City (Queens) and Brooklyn Downtown. Both of them also have easy subway access into Manhattan. Brooklyn and Queens are also interesting boroughs with good restaurants that may be less expensive than Manhattan.

Distilling it down, you need to determine what tourist attractions you plan on visiting (or at least the top ones you're most likely to visit). If they are in different parts of NYC then where you stay matters less as long as you have good subway access. If they are all in one area then you could pick a hotel in that zone or still stay further away with good subway access to reduce nightly room costs.

One more example, if you're going to do a museum (American Museum of Natural History would be good with younger kids), Central Park Zoo, a broadway show or The Rockettes Christmas Spectacular (highly recommended), The Statue of Liberty, and Chinatown, they are all in different parts of Manhattan. So no matter where you stay you will either be doing subway/bus or uber/taxi to get to many of them.

--Jon