Hotel in Rome or Neighbourhood

That is a pretty wide open question. Ten responses will give ten different locations. We prefer the Termini area for its convenience to transit and local restaurants. Some don't like it because they think it is gritty. But a lot of Europe is gritty. We prefer Hotel Sonya, Aberdeen, and sometimes Contilla if it is just a one night stay.

Somewhere in the area bounded by Campo dei Fiori, Piazza Navona and the Pantheon. Very walkable and good access to buses as needed. We also like Prati, near the Vatican, with a residential feel and excellent transportation options. Get an apartment by all means. Check VRBO.com and Booking.com for starters.

Looks like your trip has changed a bit.
Makes more sense to start in Rome if the girls want to see Pisa.

Try http://www.cross-pollinate.com
For apartments in the neighborhoods
Laurel has recommended. Just make sure they have AC.

You cannot beat the:
Piazza Navona and the Pantheon areas for easy access to the sites and in general the perfect location if you want to stay in the heart of the historic area.
Look for something as close to a piazza as you can find.

It is not the most expensive area to stay but far from the least expensive so cost may dictate your choice. If not I would say this area is better than any higher priced area. You also likely have to account for a little higher transportation cost with luggage for this location.
You can save money by staying in the Termini area and also be closer to the train station for those 2 reasons it is a very popular choice.

My favorite hotel in Rome is the Hotel Smeraldo near Campo de' Fiori. The location is great and the rooms very reasonably priced. It gets you away from the gritty area around Termini station and major attractions are all within a reasonable walking distance. I've stayed in their Classic and Superior rooms, and for the few extra Euros would recommend the latter for sure (so much that my fiancee and I are staying there for our honeymoon this fall). http://www.smeraldoroma.com/en/

I will toss my hat in the ring on this question. I stay at hotel EXE della Torre Argentina. It's very close to hotel Smearaldo recommended above. I have not stayed at Smearaldo. Either way, the location is great for these two hotels.

Moose,

I also prefer to stay in the area near Termini station, as it's so very convenient for transportation. I tend to use either Hotel Sonya or Hotel Aberdeen.