Chinatown Ice Cream Factory! (down the block from the restaurant I mentioned above)
Lion King might be OK, depends on the 16 year old. (note: I saw Rent with my D on her 15th birthday, LK would not have been her cup of tea).
I think @Novacat9191 has some good suggestions.
If your 16 year old is into art, and you feel like venturing to Brooklyn, we LOVED going to Bushwick to look at street art…something outdoors, and different!
Sorry- I didn’t see the evening part. That would be more of a day activity
Can’t get more authentic than Big Wong (no joke) on Mott Street. I think they are reopening soon.
Go for a bike ride. Lots of bike shops rent out. The parks are closed to traffic on weekends (?). You can bike safely along most of the length of Manhattan along the Hudson River - Hudson River Park and Morningside Park.
The Rockaways have great public beaches. With the water at its warmest, September is the best time to go to the beach. Pack a picnic lunch from a NY deli and spend the day. Or go to the beach in Brooklyn at Coney Island and ride a roller coaster while you’re there. Visit the nearby Aquarium.
Bronx Zoo is spectacular at any age.
There’s a ferry that goes to Rockaway Beach from Wall St. $2.75 each way (pegged to the price of a subway/bus ride).
We did enjoy the boat ride around NY. We also enjoyed high tea (but daytime activities). Both teen kids liked the munchies included with the tea. We were at a very nice department store, right after boat tour.
If you want to bike you can get a citibike but I would definitely not do that at night unless it is on a closed street.
There are now many protected bike paths, especially in Manhattan, where it is relatively safe to use Citibikes. They are not really good for long-distance biking–very heavy and clunky, with the need to swap out after 30 minutes unless you have a longer term membership in which case it increases to 45. Renting a more appropriate bike for distance riding may be a good idea. Above I mentioned Governor’s Island, which used to be a Coast Guard base and is now a public park accesbile by a very short ferry ride from Brooklyn Bridge Park or Wall St. pier. There are bikes to rent there. I am not sure when they close for the season though. https://www.govisland.com/
I wouldn’t do it an any street.
I was going to suggest biking on The Highline (never visited myself, just on my wishlist). But I see it is only for walkers. Walking would be fun too (this link talks about biking to it) - The High Line | Citi Bike NYC | Citi Bike NYC
Blue Man Group is fun to see and I think it’s running.
Lots of great suggestions–just adding that a visit to a NY Deli might be fun, especially if he’s not been to one. We like Katz Deli (205 E. Houston).
I second the deli idea. We like Second Avenue Deli (2 locations in Manhattan, neither on second avenue).