Watertown NY is over 300 miles north of the city , I'm not sure there is any form of transport that would be feasible for a day trip
If they can spend a few of days, they could drive to a commuter train station north of the city and do a visit from there. http://web.mta.info/mnr/html/mnrmap.htm (There are also Amtrak trains from farther north) or park in New Jersey and take a train, bus or ferry into the city
Not sure if there is any suitable bus service from upstate.
In order to get a bus, you need to head to or at least through Syracuse I believe but the bus will take most of the day so you have to stay overnight. The train is out of Syracuse as well but it's not that much faster if at all as a bus. I have driven from Syracuse to NYC for 2 shows & a buffet lunch (at a place that has long since gone out of business) in one day with a friend but we were young & foolish & exhausted by the time we got home & then you'd have the drive to Watertown. The only way this could possibly work is to take the 1st plane out of Syracuse (again you'll have to drive to Syracuse from Watertown (which, if I remember is about a hour drive) and then take the last plane back (again with a drive back to Watertown.) If the uncle wants to treat them, I also say they should stay at least one night. I'd suggest that Ottawa Canada is a much more doable day trip but the uncle & kids need proper documentation to go into Canada.Edited: 12 June 2018, 21:38
Honestly, doing a day trip, especially without driving yourself, really is not feasible. How old are the kiddies? This would be an extremely long and tiring day for them. They may not enjoy themselves if you make this a 14 hour day for them. Any way they can stay overnight?
I don't see any realistic way to do a day trip from Watertown NY unless they fly, but given the cost, they could drive and stay in a nice hotel in NYC for a night or two. (FYI: The RT flight from Syracuse to any of the NYC airports plus ground transport is ~$300 to $350 per person.)
If they can stay overnight, then drive from Watertown to Albany (~4 hrs) or Poughkeepsie (~5.5 hrs) and either (a) stay overnight and take an early train into Manhattan the next day (2.5 hr from Albany; 1:35 hr from Poughkeepsie); or (b) take the train into Manhattan, arrive early afternoon, stay overnight, and depart the next afternoon day.
I don't recommend taking the Amtrak from Syracuse (~6 hrs) or Utica (5 hrs) because the service rarely runs on or close to schedule for trains that originate west or north of Albany.