We stay at hotel Rosetta on via Cavour. It's a lovely, quite basic, family run hotel next to the colosseum. Rome has seriously some of the worst hotels I've ever been, so do go somewhere that comes with good reviews. Have a lovely time, it's an amazing place to visit.
Top tip (don't know about kids) is that throughout the centre there are pipes with lovely cold free drinking water, so bring a bottle and fill it up rather than pay extortionate street prices. The pipes also have a hole near the end. If you put your finger on the end of the pipe, it forces the water up through the hole and you can use it as a drinking fountain!
We stayed in a great apartment - can't find the details as my friend booked it. Not sure if it has a/c but shutters and thick walls so hopefully would be ok in summer, great location just off a great tourist square, whose name I can't remember . Do you want me to find the details?
It says 3 bedrooms but actually it is 2 bed with a cubby bed on top of the second bedroom.
Loads of room, well equipped kitchen including cling film and foil,free wifi and not expensive. Lovely location, minutes away from the Piazza Nevona (sp?) with lots of inexpensive restaurants within a couple of minutes walk. Reasonably quiet as not on the main road.