We stayed in a wonderful 2 bed/2 bath apartment less than a block north of Pizza Navona. It was definitely cheaper than two moderately-priced hotel rooms and much more comfortable.
There are several 'holiday apartment' web sites out there and I've found that most apartments are listed on most of them. I found the rental experience to be very easy and rewarding. (I booked my Venice, Florence and Rome apartments via Trip Adviser and paid via paypal.)
I would stay in the general area around Piazza Navona. This puts almost everything within a reasonable walking distance for most people. Although, if booked on an early morning tour of the Vatican, a taxi might be advisable (but it's only about e10.)
A Rome hint - If arriving by train, only get into a licensed taxi. Before getting in, check for taxi license stickers and a functioning meter. If you don't see these, decline and wait for the next one.