Jane Addams Hull House Museum (http://www.hullhousemuseum.org/) in Chicago on the UIC campus.
The La Brea Tar Pits outside Los Angeles - lots of sabre tooth tiger and mammoth skeletons. It’s small but fascinating.
RIP the Higgins Armory Museum in Worcester MA. Happily, it seems the collection is finding a new home: http://www.worcesterart.org/director/higgins-collection/
My younger son loved the Armory Museum.
MIne did, too!
Presidential libraries? The JFK one in Austin is great, as is Clinton’s in LR.
And while you are in Austin, drop by the UT museums, the natural history one and the Blanton Modern Art one. Don’t miss the Bob Bullock Texas History Museum right across the street from campus. If you want to understand Texas and how Texas sees itself, this is the place.
If in Memphis, go to Graceland! There are tours and transportation from the airport if you just have a short lay over.
When in Vegas, drive out to the Clark County Museum. Great history of the area, including gambling and mining, with lots of historic structures saved from around the region. A great non-commercial, inexpensive experience for families. There is a cut-out of the “Beard of Knowledge” from Pawn Stars, as this is where he works.
When in Chicago, go the Museum of Science and Industry in the Hyde Park neighborhood and plan to devote the entire day since this is the largest public science museum in the western hemisphere. This is on the south side, away from the more well-known museum campus that includes the Shedd Aquarium, Field Museum and Adler Planetarium.
When in Madrid, Spain, the Maritime Museum is free (requested donation) and the paintings, ship parts and gun collections are amazing.
@hanna you asked
The Merseyside Martime Museum in Liverpool does. It is a fantastic museum overall but it was incredible to be able to see and even lie down in the spaces just like my grandparents came over in.
My favorite stumbled upon museum is the Advertising Museum in Tokyo. I don’t think my family ever enjoyed a museum more. And it was free.
In the slightly more accessible category, although not a museum, I recommend the Franciscan Monestery of the Holy Land in America in D.C. They have fantastic gardens, and replicas of various shrines in Jerusalem. They also have replica catacombs that you can tour. Having also been to the real places I can attest to the fact that they do a good job.
I have not noticed anyone mentioning my favorite - THE SPAM MUSEUM in Austin, MN. It’s right off the interstate and uses cool Burma-Shave type signs to lure you in. It’s a ton of fun.
http://www.spam.com/museum
I don’t know if I would classify this as a Hidden Gem, but more a legendary family experience. Disclaimer: this was around ten years ago, so things may have changed.
We were on vacation in Grand Lake, Oklahoma, and visited the Har-Ber Village Museum in Grove, OK.
It started out okay, but it got weirder and weirder as we went. Some of it seemed to be collections of stuff people didn’t know what to do with any more so they donated it to the museum.
Our kids were younger at the time (maybe junior high and high school?) and we were reduced to fits of the giggles.
I don’t know what was better: the creepy clown collection, the salt & pepper shakers, or the display of a brick from every state.
Unfortunately, the best exhibit is no longer on display. There was a display of hand-made Christmas ornaments from a woman who illustrated some of the scenes not typically depicted at the holidays - or anytime, for that matter. The museum link below includes another link to a blogger who visited the museum and captured some of the more “interesting” ornaments. If you are looking for a new idea for a themed tree, here’s some inspiration:
I visited Case Western last week and stopped by its Dittrick Museum of Medical History. I had just 45 minutes before it closed, but it blew my mind so much that I came back the next day. Remarkable artifacts covering infectious disease, surgery, childbirth, the development of technology like microscopes and X-rays, and a whole room about millennia of birth control from the Pharaohs to the present.
Talk about a hidden gem. And it’s FREE! If you’re in NE Ohio, don’t miss it.
Well, those ornaments from the Har-Ber village are …different. I’m not sure which I would put front and center on my tree: Paul and Silas Whipped in the Marketplace, or the Hanging of Judas. (or Absalom’s Hanging, which is a different take on the subject.) Christmas must have been a happy, happy time in that household. Eek!
http://secretfunspot.blogspot.com/2008/12/old-testament-christmas.html
Visited the Frick in Pittsburgh today. It is free to walk the grounds and see the permanent collections , Car and Carriage Museum (great), and Greenhouse… There is currently an exhibition of works by Monet, Degas, and Van Gogh that there is a charge for( but it is well worth it) and there is also a charge to tour the Frick home, Clayton, which is on the grounds .
UMich’s Muesum of Umich Football was surprisingly Interesting. Especially the exhibit about President Ford ( man that guy was good looking when young) and his teammate who he roomed with Willis Ward ditto). When A southern school threatened to forefit if Michigan played the African American Ward , Ford threatened to refuse to play if Michigan acceded, only to be persuaded by Ward to play. Ward ( who became an auto industry exec) and Ford remained friends into old age.
Longwood Gardens, the former weekend and summer home of Pierre Dupont, is definitely worth a visit. I went for the first time recently and would definitely go back. Easy day trip from Philly. Really spectacular. A volunteer told me they have 300 paid employees and over 500 volunteers which is an indication of how extensive the gardens and property are.
@Hanna when I was young the Medical Museum on the Mall used to be my favorite place. It was pretty gruesome - pickled body parts, a whole exhibit on syphilis. But we kept dragging our parents back there to their dismay. It was moved in order to build the Hirshorn. For a long time it was at the Walter Reed Army Hospital, but it looks like it is now in Silver Spring, MD. http://www.medicalmuseum.mil/index.cfm?p=about.index It’s free too. 
I visited Woodlawn and the Pope-Leighey House today, south of DC. Very interesting and the Frank Lloyd Wright house has a connection to Loren Pope , of “Colleges That Change Lives” fame. http://www.woodlawnpopeleighey.org
Friday, the 18th, is International Museum Day. Some museums are offering free admission.
And the museum I started this thread with, The Mariners’ Museum, is offering, $1 summer admission, May 25- Labor Day . http://www.marinersmuseum.org
The Williams College Museum of Art, with the wonderful eyeballs on the lawn.
In London, I love visiting the Cabinet War Rooms, where Churchill’s bunker was.