We went in October with our 11 year old but I’ve been previously in a December and January & it snowed both times. It’s bloody cold obviously but looks very pretty in the snow - just layer up!
In terms of what my DD liked (or didn’t!), if you are very familiar with the Natural History Museum in London, the NY one will be a bit of a let down (though it’s a nice building). She’s also into singing & dancing - we booked a Broadway show through TodayTix (download the app, they offer discounted theatre tickets bookable in advance. Switch the location to NY. Their rep meets you outside the theatre with the tickets). If you do a show, beware of the incredibly silly bar prices! She/we also liked Times Square at night - previously it’s somewhere I’d walked through as quickly as possible but at night there’s something Blade Runner about it!
Other than that, we all liked the High Line & walking Brooklyn Bridge. We did go to the Friends building at DDs request but I really wouldn’t bother!
We didn’t go in the 9/11 museum but did visit the site which is worth visiting and is moving. It’s not far from the ferry terminal to Staten Island which is a free and easy way to see the Liberty Statue.
Plan your trip in zones and as you’ll be queuing for things, dress for that. We did Top of the Rock as it was marginally cheaper than the Empire State but be aware even if you buy an advance ticket, unless it’s a VIP one you’ll get a time slot when you get there and have a couple of hours wait time. The Empire State seemed to have less of a queue when we passed it.
Other than that, DD got v excited by all the crappy doughnut and fast food options and Sears department store - she’s only 11 so got time to develop some taste!