@Mosschopz
It’s not the same but I took my 6 year old three years ago. We got a hotel with a pool (the Dominick in SoHo) so however boring the day was for her and it never was!) she could come back to her own fun. Massive age difference but that was what we did. She cooed WAY better with jet lag than we did too.
Yes - this is what we used to do when we travelled with ours at that age. We live in the US and we fairly regularly took them to NY, Boston, Chicago, etc. Having a hotel with a pool meant they always had something fun to look forward to.
OP - I'd wait till he's a bit older, only because I found it so much fun taking kids to NY once they were 7 or 8, and able to walk longish distances fairly reliably.
We loved the Cooper Hewitt Design Museum especially - it has a garden, so you can head out there periodically and recharge, then go back in, and there's loads to see.
I also took mine to a Broadway show, we ate out lots of different places, walked the High Line, biked round Central Park, did Top of the Rock, American Girl doll shop, did a night-time bus tour, admired the shop windows on 5th Ave, went to Chelsea Market and Gansevoort Market, had lunch in a bar in Grand Central Station, visited the Apple store in Chelsea.
We regularly got bagels or breakfast sandwiches from the local bodega.
The one area my kids hated was Times Square. Way too crowded.
We didn't manage to fit in the Museum of Natural History but if you do make it there then read The Rosie Project first.
I also love Boston for kids - there are several great museums including the Science Museum, the Children's Museum, and the Isabella Stewart Gardner Museum (although it's not aimed at kids, it's an amazing place).
It's also fun to take them to a game - ice hockey is great, basketball is also fun, baseball is loooooong.