Thanks to all for your advice
I had been trying to book a room for 4 days. We are getting into Memphis at 11:10 PM. I did not want to drive too far since we are arriving so late. As some had suggested, I tried calling the hotels directly. The Peabody put me on a wait list, but it did not sound too promising. Starwood was no help at all despite my humble Gold Status. The Westin would not even put me on a wait list.
It was recommended I try a travel agent. I called Amex Travel. Again no luck. Marriott offered me a Courtyard room but with a 3 night minimum stay at $189 per night. That is almost $600 for 2 nights. I kindly explained those were Ritz Carlton rates. The representative was understanding and laughed.
I finally tried Kayak. I had not used this website before. Suddenly a the Quality Inn at 6068 Macon Cove , Memphis, TN showed up with one room left. It was $152 per night. Steepest price for a cheap hotel I have ever paid. It is however only 6 miles from Rhodes College where my son is interviewing. I went ahead and booked it. I feel fortunate to have found a room even at this rate given the situation.
Again, thanks to all for your advice. Persistence and a little luck paid off. The recommendations will come in useful for the future in finding a hotel when a town is packed for a convention.