Any hotel near Time Square will have good subway access which I think is the most important factor here. My two favorite IHG hotels in NYC are Kimpton Muse (46th between 6th Ave and Time Square), and Intercontinental Time Square (actually in Hells Kitchen 44th and 8th Ave). Both of these properties are right on top of subway stations so they are very easy to get to/from major tourist attractions in other parts of the city.

For first time visitors like your kids, I really think they should stay in Manhattan to get the real feel of the city. And the best place in my opinion is Midtown, not necessarily Time Square but anywhere nearby will do. Financial District is fine if you are on a business trip but it is not 24/7 like Time Square... seeing the sea of people walking around the street at midnight around Time Square from your hotel window is part of the NYC experience.

Food options depends on your kids' palette... I suspect you will end up in Little Italy and Chinatown eating subpar pizza and merely adequate dim sum at tourist trap places anyway - again... part of the experience, not passing judgement Yelp is your friend here.