Trying to decide on which neighborhood to stay in, in Rome

We just stayed in the Monti area (near Colosseum and Forum) for the first time -- lots of youthful activity, restaurants, easy walk to sights. Had a flat (Boutique Grifone) which does have a room for 4.

To be closest to sights you should stay in the historical center (Trevi, Pantheon, Navona, Spagna, Campo dei Fiori).
Trastevere, Vatican, Monti, Via Veneto, Prati, also good.
Check Homeaway (VRBO) for apartments.

Have you plotted the top "to see"s on your list? Once I did I decided that P. Navona was the correct area for me. We would be close to our top items plus near bus stops. Spend some time with Google maps - I think it'll help.
Also, knowing how teenage boys eat (and eat and eat) I would recommend you consider an apartment. We found one with a full, albeit small, kitchen using Homeaway. We could have cooked a meal with ingredients from one of the lovely markets but we didn't. We did, however, enjoy breakfasts and a couple of takeaways there.

Roberto,

I certainly do plan on staying in the historic Rome area. But what I am looking for is what people think of various neighborhoods - what do they like (or not) about them. For example, if you compare the Monti and Trastavere neighborhoods, what are your opinions?

We intend to spend a day (or chunk of it) at the Vatican, and a day walking around the Forum/Coliseum area. With essentially another day and a half, we will walk around various areas - I expect we will go to the Pantheon, Trevi Fountain, etc. I also want to go to the Borghese. We will likely do at least one of Rick's suggested walks.

I have already had several people mention the Piazza Navone area as a favorite, and I will definitely e looking around for places there

Trastevere is a great area, lots of fun restaurants and a good area for wandering. There's also plenty of younger people around so your kids will enjoy it. Only drawback is that it is just a tad farther out than the Campo de Fiori or Navona area. Those would be best in terms of location and are also great walking areas.

Monti is nice too, and great access to Colosseum/Forum. It's less convenient for Vatican or Pantheon etc - what is your priority?

My first visit as an adult I stayed on Piazzs Farnese which was ideal - great access to everything. Less noisy than Campo de Fiori but right next door so you get the market and feel of that area - teens would like it.

The way I like to explain or maybe justify it since budget wise it may not be the advise you want to hear.
This is personal opinion of course.

There are many areas you can stay in, many different areas you will want to see/spend some time in but only one in which you would want on a single trip to return to multiple times.
That area is the historical center and think the primo location for that is the Pantheon area.
The closest option to that which often has more availability and maybe better values is Piazza Navona area.
Trevi area would also be good but unlikely any values to be found there.
A little further but still convenient is the Campo Fiori area is the next closest to the center that is likely to have some good deals and availability.

Any other areas to me are places I want to go to once and will make the rest of the time spent less convenient.

For me, Monti was great - lots of character, plenty of restaurants around and you can walk to the Forum and Colosseum within 5-10 minutes. This saved us a lot of time and hassle. If you don't mind walking, it's not even that far to Pantheon/Trevi Fountain/Piazza Navona (30 mins). There is also a subway station and plenty of bus routes that will take you everywhere else. We visited the Vatican and it was an easy (and very cheap) ride on the Metro, ca 20 minutes.

Also a 1 minute subway ride from Termini station - great for airport transfers.