If the weather sucks, that is the perfect time to hit a museum. NYC has dozens, from an immigrant museum on the Lower East Side to a stop on the Underground Railroad in Flushing, in addition to all those huge, world famous ones like American Museum of Natural History and Metropolitan Museum of Art
The best thing to see in NYC are the little details of whichever neighborhood you find yourself in. Skip the big, obvious tourist crap and just enjoy the local flavor. Bring an umbrella.