Seems like you have more than enough opinions on the train versus driving, etc., so nothing to add there (other than saying Mahwah seems like the worse option).
It does seem like a ton of trouble to have come all that way to the most iconic city in the country during the most iconic season and rushing in and out just long enough to see a show. I agree with others – meet the relatives at the show but go earlier and at least take in a few sites, all easily within range of where Hamilton’s theater is. Of course you’ll see Times Square (because its near the theater and you walk by from the train station) but I wouldn’t hang out there longer than the photo op – its actually the worst place in NYC. But the winter market at Bryant Park just a block east would be perfect, and right next to the iconic main library. Or the ice rink and tree at Rockefeller Center, just a few blocks north of that.
Or if you’re looking for something totally different and nice and warm inside, right next to your train station (Grand Central), you could go to The Summit – the 3 story viewing “experience” at the top of one of NYC’s newest skyscrapers, where you are higher than the Empire State Building. I’ve done many famous top of building viewing experiences, and this one was an order-of-magnitude better than the rest. You can see all of NYC, but that’s honestly the least interesting thing about it. The floor is a mirror and the whole thing is surreal.
There’s obviously a thousand other things to do in NYC but I was just listing a few that could be done in a couple hours near where. you will already be.
(side note, if you’re planning a sit down lunch or dinner, the restaurants very near Times Sqaure and the theater district are generally a worse experience than the rest of the city – way too crowded and usually not great food in what is otherwise a city with over 10,000 places to get great food.)