Except we never closed Gitmo, and even if we did, the public outcry is associated with that base's name alone. If the military were to rename and move it a couple miles over, the public perception wouldn't ever reach Gitmo levels at least as long as Obama is in office.
And we did it before with the School of the Americas.
It was in Panama as the "U.S. Army Caribbean Training Center" and was forced to move as a result of the Panama Canal treaty. Then they changed the name to School of the America's and focused on fighting communists (with the same tactic of training what we now call insurgents).
After the killing of nuns in South America it started to get bad press (referred to as the "School of Assassins") and eventually was forced to "close" and reopen as "Western Hemisphere Institute for Security Cooperation".
Gitmo seems like it is about to undergo a similar process.
Edit: Just saw that today is the 30th anniversary of the assassination of Archbishop Oscar Arnulfo Romero. The people responsible were trained at the School of Americas.