I recommend the cemeteries, especially Green-Wood and Woodlawn. Unfortunately the first is in Brooklyn and the latter in the Bronx.
Hundreds of enjoyable tombs in each, though. Those rich people really know how to live.
I enjoyed it also, but you need to be interested in the period. My father grew up in that neighborhood, so it had special meaning to me.
A visitor should be aware that there is no museum per se (though there is a good gift shop) and you attend by going on a tour.
I have been to the Brooklyn museum a number of times and it is quite interesting. The most notable thing is Judy Chicago’s Dinner Party: https://www.brooklynmuseum.org/exhibitions/dinner_party, but there’s other good stuff. Another off-beat attraction I have been to several times in the Transit Museum also in Brooklyn not far from the Brooklyn. Once I stumbled into something called The Old Stone House in a park bounded by 4th and 5th Aves. and 3rd and 4th Sts. in Brooklyn. It houses a small museum devoted to the story of George Washington’s army in Brooklyn and their narrow escape. His entire army could have been captured and the war ended there.
one of my earliest tourist memories was 64/65 NY World’s fair. I mainly remember Mom got her wallet with credit cards stolen and she had to eat Belgian Waffles. 
The horror! I believe the Belgian Pavilion and its waffles were where the concept of fancy “desert” waffles first was popularized in the US (as opposed to regular breakfast waffles). In any case, it was the first place I had such waffles.
As mentioned above, this is at the old World’s Fair grounds, which still contains a few of the other original buildings (the Queens Museum is the former New York City Pavilion, first built for the 1939-1940 World’s Fair), including the Unisphere and the New York Hall of Science.
You can go see the old fire station used in the movie Ghostbusters.
You did say Manhattan, but if you get down to Brooklyn I recommend the Transit Museum.
Add me to the people who enjoyed the Tenement Museum. A few of their tours, and the one I did, are “experience” tours, where an actress plays the role of an immigrant who lived there ~100 years ago.