Many good options have been given. Broadway Plaza is a great place, but they do not allow pets. The Berkman is a hotel/apartment that is across from the Saint Marys campus. They are pet friendly, have multi-room setups and have a shuttle that can bring you to the Downtown campuses. They will have discounted rates for Mayo Clinic guests as well. I will include their information below.

The Berkman
217 14th Avenue SW
Rochester, MN 55902
(507) 237-5626
http://www.theberkmanshorttermstays.com