The Smithsonian's Air and Space museum had a huge Trek exhibit about 5-7
years ago that was a big success.  Their Star Wars exhibit, which opens in
November or there abouts, is planned along the same lines.

From what I can recall, it had a bunch of props, set pieces, photos and
memorabilia from the old series and a mock-up Captain's chair that you could
sit in and take pictures of.  I don't think they charged for the exhibit but
you did have to get tickets because the lines were long.

You could probably write of call the Air and Space museum in DC and see if
they could give you any pointers.

Deb Fuller