Names of hotels near termini for overnight stay before heading out to the airport.
For our quick overnights we use Hotel Contilla. One block south of the side door on the south side of Termini. It is a 3 star so not sure if I would classify it as budget but it is handy.
We booked one through the website noted below. 29 Euro per person. We've yet to go but it looks decent enough and it is about 10 minutes walking distance from Termini. http://cross-pollinate.com/info.asp?id=287&kind=2&city=rome
I like Hotel Sonya, across the street from the Opera house.
Hotel Des Artistes Via Villafranca, 20 Rome 00185 (Italy)
+39 06 4454365 We stayed in a double for around 70 - 75 euro per night.