I asked my Italian colleague Piervittorio, and here was his advice...
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Try these web sites --
http://www2.italy-hotel.com/category...6&category=983
http://www.virgilio.it/servizi/viaggi/008.html
http://it.dir.yahoo.com/Aree_geograf...sidence/Hotel/
My advice... try not to look like an American, because the prices will magically grow!
It is fairly easy to get around Rome (via buses, metro, cabs, etc.), so in my opinion it is better not to stay right in the center because it is more expensive indeed.
$200 is actually a reasonable expectation, and that in itself can take you pretty far. I would recommend staying in Trastevere (more characteristic and less touristy!!), or perhaps look for a little Pensione. If you like the Spanish Steps (in Italian -- "Piazza di Spagna") though, look into those web sites. In September price should go down a bit!
I'd also recommend staying around Piazza Navona / Campo dei Fiori -- it's a much hipper area!