I don’t mind living in NYC in the winter (I actually prefer living there then… I’m from MI) but for touring it seems like mid-January could be a little miserable. Crowds are negligible (for NYC, at least) but that’s usually the coldest part of the winter so you WILL need warm, weather-proof boots, heavy duty winter coats, hats, gloves, and scarves because you’ll be outdoors a lot of the time. Even if you plan mostly indoor things, you’ll be walking a lot unless you plan on taking taxis to get everywhere. 15-20° is cold for everyone, but it can be absolutely brutal if you’ve been south all winter! There may or may not be snow… in '12 and '13 we had close to zero snow in January, '14 we had a bunch, '15 it was really cold but we only had a moderate amount of snow (and mostly at the end of the month), and then this winter it was mostly warm in January except for a HUGE storm at the end of the month… close to 2 ', but then it was 50° a week later. February tends to be a snowier month in general, but it’s warmer then so it melts faster.
That said, I’d recommend a BigBus tour. You can buy the tickets online, but you can often get a better rate if you buy them from the rep that’s at the stop in Times Square. It will stop at pretty much all of the major tourist attractions (with a live tour guide who tells you about all of the stuff you’re passing), and you can buy combo tickets that include things like the Statue of Liberty, One World Observatory (the new World Trade center), Empire State Building, and a few other ticket attractions. I’ve actually taken it myself, just because I wanted to be a tourist for a weekend!
As far as accommodations go I’d recommend the Times Square/Theater District area, (Central Park south area as a second choice… slightly further away from the things you’ll be wanting to do, and in the winter you’re not likely to spend a lot of time in the park), simply because it’s the one place where you can get on just about any subway line, and all of the tour buses stop there. Plenty of places to eat, and lots of hotel options, as well as being close to shopping, theaters, and lots of other attractions. My personal favorite hotels there are the Crowne Plaza Times Square and the Intercontinental Times Square, or Park Central Hotel in the Central Park south area. If you want to book through Hotwire Hot Rates (where you don’t know which hotel until after you book) then I’d say you’ll be safe with anything 4-star and higher in the Times Square South and Central Park South areas. The one cruddy 4-star I’ve ever stayed at in Manhattan (The Manhattan at Times Square hotel… avoid at all costs) is in Times Square North.
Kids usually love Ellen’s Stardust Diner (51st and Broadway) and it’s worth the wait… the waitstaff take turns singing, and they’re usually really good. Totto Ramen is nearby and it’s one of the best ramen places in the city. Also, if you’re big breakfast eaters then finding delivery on seamless.com is going to be a lot cheaper than hotel breakfast and you’ll find lots of options there at any time of day.
He wants his NYC trip, and SNOW! (Yea from me, truly, being a FL resident, I miss cold. Like, REALLY miss it! 
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I got one picture in Central Park of my 10YO kneeling next to a dirty, melty, 12 inch square gob of snow that hadn’t yet done itself in. I’m hoping for better Elsa-like snow effects if I take Thing 2 earlier in the year.
