I somehow missed a story on Slate (by Hugh Ryan) in October 2013, but better late than never. There has been talk of building an LGBT History Museum as part of the Smithsonian, perhaps toward the eastern (Capitol) end of the Mall in Washington. Slate also suggests what a lot of the content would he.
There are numerous museums or gay history collections around the nation, with the largest now being in Washington, New York, Ft. Lauderdale, and San Francisco. I will be contacting them soon with information about my books and will have more details soon on one of my Wordpress blogs.
Update: September 10, 2016
Slate has suddenly deleted its story (it gets a 404 even when searched within its own site). It seems that interest in a museum on the DC Mall has waned and the new idea is to build one in New York City. Washington Business has a story by Rebecca Cooper (paywall) here. It seems like the New York Mets won this one, metaphorically ("Washington at New York" in MLB parlance). Stay tuned on this one; more could change.