Best Neighboorhood to stay in Rome for family of 4

You seem to have changed your wants/needs quite a bit.
Not sure this question is phrased to really meet your wants?
With this question worded as it is I suspect you will get a response of near the Termini Station but based on your earlier question not the area I would lean you toward.

You will be there in 12 days and you have no hotel reservation?? Yikes....The best neighborhood to me is Monti, close walking distance to Victor Emmanuel area and the Colosseum lots of cafe's and places to eat, plus it is a very old neighborhood. We saw lots of families walking around. You might start by looking on booking.com and reading reviews or on Trip advisor.

At this late date your best option is either booking.com or Venere.com. For all cities listed. You might check out their apartments also. Much more room and less expensive.

Along with Monte, Navona, Campo Di Fiore and Pantheon
Neighborhoods.

Hotels in Italy book by person not by room. Make sure it states you have A/C, many do not.

Good luck in your search and have a great trip.

What is your budget and what are you looking for? Rome can be very expensive.

In the same vein, we are considering Rome for a milestone birthday next February or March. 8-9 days in a non-touristy 'livable' neighborhood. Have been looking at the area just north and slightly east of Termini, near the British Embassy. Might take a daytrip or two, maybe catch up on a couple attractions we didn't see last time around (only 2+ days in 2010) but basically want to 'do as the Romans do'.

Jay,
You will get better and more personalized responses to your needs if you post a separate question.