Londoner here and repeat NYC tourist. These are three you might otherwise miss.
  1. The Tenement Museum in the Lower East Side
    Oh how I love this place. Educates you on the immigrant experience over the city's evolution but does so by telling the stories of the people who once lived in the tenement building. Storytelling at its best. Also ace gift shop.
  2. The Ukrainian Museum
    A weird and wonderful guidebook directed me here a few years ago. I was the sole visitor for the hour I was there and they assumed I was Ukrainian. They treated me like a queen and made me sign up to their mailing list where I receive on average 2 emails a week about Ukrainian happenings in the city.
  3. El museo del barrio
    Wonderful find not too far from Central Park, devoted to 'Latino culture in the Americas'. Well worth a visit.