If you're okay with camping, I stayed at Camping Tiber this October for 5 days, as part of a 31-night 1000-mile long self supported cycling trip across the country. It's located at Prima Porta, a ten-minute commuter train ride from the Flamino metro station, at Piazza Populo, a very convenient place to start your explorations of Rome. In terms of accommodations, they had a pleasant shaded area for tents, and the nicest, cleanest, newest-looking bathroom/shower house of any that I saw across the entire country (and I stayed at around 20 different campsites). For about $20.00 a night (this was for 1 person+tent, yours will be more but still reasonable), it couldn't be beat.