Rename the install folder, keep the same URL

I think I’ve read just about every FAQ on the subject and something still isn’t right. Perhaps I’m over-thinking it. Here is what I would like to do.

The URL will stay the same, such as http://www.mysite.com
Rename the host subdirectory from say http://www.myhost.com/red to http://www.myhost.com/blue

Any information would be appreciated, thank you.