My university, Miami(OH), took their name from the Miami tribe that lived in the Southwest part of the state. Their nickname was the Redskins, and was changed in '97-'98 at the request of the Miami Native Americans, to the Redhawks.
The school has been around since 1809, so much longer than the Washington football team. I think they, and their fans, will get over the name change.
My baseball team in Cleveland is most likely changing their name also, and the people complaining about it baffle me. It's a nickname/mascot of a sports team, there are bigger fish to fry in this life.