You can spend the night in Memphis at a clean, safe and well-lighted place for $15 per night, and the only catch is that you've got to do a chore or two. Assignment #105 in the 365 Things to Do in Memphis is to stay at the Pilgrim House Hostel.
The Pilgrim House, which is the only proper hostel in Memphis, is located inside the First Congregational Church building in Cooper-Young. Their rooms start at $15 per night for a shared room. There's more info on the Pilgrim House right here.
1000 South Cooper Street
Memphis, TN 38104