The American Visionary Museum in Baltimore and the Secret Gardens in Philadelphia have funky art installations that use a lot of glass, mirrors, and found objects.
The John Michael Koehler Museum in Sheboygan, WI celebrates found art, and the self taught artist.
If you visit the DeCordova Museum in Lincoln, MA, you should see/tour the Gropius House, which is also in Lincoln. The house was designed by Walter Gropius, the German architect who was the founder of the Bauhaus movement. Gropius built the house for his family in the 1930s when he came to teach at the Harvard University School of Design.
Also, I don’t think anyone has mentioned the Isabella Stuart Gardner Museum in Boston–not exactly unknown, but it’s a definitely a gem. Gardner and her husband Jack, wealthy Bostonians, began collecting early in their marriage. Eventually their collection grew so large that they decided to build a Venetian-style palace/museum to house it. Mrs Gardner, who was always shocking Boston society (she once wore a head band to a Boston Symphony concert that read–Go you Red Sox) ended up building the museum on her own when her husband died suddenly.
Since I was little, I’ve dreamed of living in the Isabella Stuart Gardner Museum building. 
The heist of art work from the museum is a fascinating story:
http://www.gardnermuseum.org/resources/theft
https://www.bostonglobe.com/metro/2017/03/18/six-theories-behind-stolen-gardner-museum-paintings/JmwHou86qo5MtBzX1fb9cI/story.html
I finally went to one of the museums I mentioned earlier. Hillwood in DC. The weather was beautiful, the docent exceptional, the cafe very nice. Beautiful gardens, nice orientation film, very interesting history of Marjorie Merriweather Post and her collections and gems. Great day and I recommend it to anyone visiting DC.
So glad you bumped this thread, @sevmom. I meant to post these earlier this month but life intervened.
In Memphis: Sun Studio, a very small but well-done national historic landmark in a still-operational studio where Johnny Cash, Elvis Presley, Roy Orbison, Carl Perkins, etc. did their early recordings. The tour guides are outstanding. It’s pretty much one good-sized room of exhibits plus the actual studio, a small eating area and smaller gift shop. I’m not especially a fan of early rock but this is so well done. http://www.sunstudio.com/
Not hidden at all in Memphis, but truly exceptional and an experience not to be missed: the National Civil Rights Museum. The Lorraine Motel is part of the complex; it’s overwhelming to stand there. http://www.civilrightsmuseum.org/
The Wallace Collection isn’t really hidden in London, but might be overlooked in favor of the better-known museums. Wonderful paintings and artifacts in a 19th-century London townhouse. http://www.wallacecollection.org/
We went to the Maine Maritime Museum in Bath several days ago with my great-nephews. It is even better than I remember: they’ve added a lot of stuff, particularly interactive things for kids.
This thread is priceless.
I second the Eric Carle Museum and the NYC Tenement Museum. Also, if you visit the Corning Museum of Glass, two the Rockwell Museum (western art) is just a few blocks away, and the Glenn H. Curtiss Aviation Museum is not far off, in Hammondsport. Both well worth seeing.
I also loved Sun Studios.
I highly recommend the World Chess Hall of Fame in St. Louis.
St. Louis has become the Chess capital of the United States in a short time. The museum is free and has rotating exhibits.
So, two men at a chess convention are bragging about their strategies and recent wins in the hotel lobby.
They are “chess nuts boasting in an open foyer.”
(Sorry, one of my FIL’s favorites puns. Couldn’t resist.)
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If you know a student in the Philly area, pass this along. From the Philadelphia Museum of Arts instagram page:
“Calling all students: now’s your chance to become a @philamuseum member for only $20. Get more art, more access, and more fun with a Student membership. Included is free general admission, multiple savings, and discounted tickets for your friends and family. Offer valid through Sept 15 in person at the Museum.”
Great deal! but just a few days left.
The Leigh Yawkey Woodson Art Museum in Wausau, Wisconsin. Every fall, it puts on the Birds in Art exhibit.
Just adding my votes for Reynolda in Winston Salem. Awesome docent led tours. I’ve been three times. And I’d go back, gardens are fabulous as wall.
And the MIM…Musical Instrument Museum…it has just about everything you can think of relating to musical instruments, kinds of music, music around the world. I’ve been there and suggested it to folks who really aren’t “musical” and everyone has raved about it. In Phoenix. I always leave wishing I could have stayed longer!
@thumper I am currently in Phoenix and just visited the MIM. An excellent museum. I heartily second your recommendation.
Aw…I put my mom in a wheelchair in the MIM, and we had a fine time together, and could not see the whole thing in many hours. What I have learned since is that their restaurant is exceptional. But don’t forget the Heard in Phoenix.
It has been a few years, but I always loved the Museum of Northern Arizona in Flagstaff, for better understanding of the cultures and geography of the Four Corners region.
The National Museum of the US Air Force, hosted by Wright Patterson AFB in Dayton, is amazing and huge. Seven airplane hangars worth of history with free admission and parking. We loved it, and need to go back soon because they are always adding to it.
The Phillips Collection (a hidden gem in DC) is doing an exhibition just about Renoir’s Luncheon of the Boating Party - http://www.phillipscollection.org/events/2017-10-07-exhibition-renoir-and-friends I have to plan a trip to DC soon!
The MIM restaurant IS excellent…but it has shorter hours than the museum.
The Columbia River Maritime Museum in Astoria, Oregon is fascinating. Piloting a ship from the ocean into the mouth of the Columbia, and then up the river to Portland, is so difficult and dangerous that a ship is required to hire one local pilot to get into the river and a second to go up the river; who knew?