In no particular order (with no attempt at being comprehensive [and not necessarily listed as an endorsement or suggestion]), you might look into the following places:
The Glenn Curtiss aviation museum, which I remember mostly as detailing the legacy of a slightly bitter dude.
Corning Museum of Glass, always good for lightening your pocket. Come for the glass, stay for the
Indian buffet.
The women's rights movement first started in upstate NY. Seneca Falls has a few interesting
museums/national historic landmarks relating to women's suffrage and otherwise telling the story.
It's the wrong season, but for two months out of the year (when the maple is flowing) there's an amazing
all-you-can-eat buckwheat pancake restaurant in Angelica.
Kodak was started/headquartered in Rochester. There's probably a good museum connected to that, but I've never been.
Western NY has its roots in a weird mix of religious minorities seeking freedom and solitude, radical leftists,
video artists, and hippies. Such an odd place.