Staying in an apartment in Rome has been a good choice for me/us both times. Hotels are generally expensive, not surprisingly, with smaller rooms. For this trip, I looked into the convent option extensively, and was really prepared to go that route. If that is something you’re interested in, a couple of good sites to start with are:
This page from the Santa Susanna website
and Monastery Stays.  The latter site is good for information about the location of various convents, although because of the added cost of the service, it might be preferable for many to book with the convents directly. I have a few regular readers who stay regularly in convents when traveling to Rome, so they can probably chime in with more information and recommendations.
The convents do vary in accomodation, some being close to hotels in services they offer, but the closer it is to a hotel in service, the closer it is to a hotel in price. Curfews are also an issue for some.
As I said, I was all set to go the convent route, except when I finally settled on a flight, it became a problem. A flight leaving at 6 am means leaving the apartment at 4 AM, a reality that was met with something less than enthusiasm by those with whom I was in contact at a couple of convents. Understandably.  So I redoubled my apartment searching, turning to Craigslist instead of VRBO and other Rome-centered listings, thinking I could probably find a better deal there.
And I did. This is the apartment in which I stayed, for a very reasonable price – so reasonable that I ended up paying, broken down on a daily basis, what I would have paid in the convents I was looking at.  And I got an apartment instead of a room. Not that I was really doing anything but sleeping there, but the privacy and room to spread out was nice.
I want to publicy enthuse on behalf of Camillia, the manager, who is fluent in English, gave a very helpful orientation, which included a wall o’ maps on which she had highlighted the bus routes and identified the locations of restaurants, groceries, internet points and so on.  She also arranged that taxi on the last, very early morning for me.

The only disadvantage for some (okay, perhaps many) will be the location.
As I mentioned, it is in Monteverde. If you look on the left-hand map in the photo, Monteverde is sort of where all the green is on the far left of the map, near the bottom. You can see the location of the apartment by the little circle with the statue in it (Garibaldi’s statue, which is the Giancolo park, nearby). The neighborhood to the east of it, bordering on the Tiber, is Trastevere. Vatican City is just North of it. Most of the other sites people want to see, from the Colisseum to the Pantheon, etc, are on the other side of the Tiber. Many people, going to Rome, want to be on that east side, in the City Center, or nearer to the Vatican. I’d say if you were going for three days or so, that would probably be preferable. But for a week, this was fine. Better than fine, to tell the truth! The Metro, of course, doesn’t come anywhere near, but the bus can be caught a block away, buses that take you directly into the Center in a matter of 10-15 minutes, depending on the time of day, or straight north to St. Peter’s.  I’m not a person to go back and forth to the apartment constantly during the day, so this was just fine for me, the neighborhood was great, with a bakery next door, an alimentari across the street, a bar on the corner (a “bar” being a place to get your coffee and light food) and lots of other shops you might need on the street. Many apartments advertise that they are in a “typical Roman neighborhood,” and this certainly was – oh, and very, very quiet at night. (In our previous apartment on Borgo Vittorio, we were right above a bar – the way you and I think of it – and it really bothered some of us. Not me, but others were kept up by the noise. Not here!)
So if you’re in the market for an apartment in Rome, and aren’t picky about location…check it out!