My husband, my 16 months old boy and I, enjoyed Rome staying at Barbara´s place. Monteverde is a very family friendly area, quite and very well connected to the city centre, just a couple of train stations from the main attractions (or 10 euro taxi from Piazza Navona).
Barbara and her husband were extremely helpful, I asked her many things by email or phone and she always replied with useful advices, she even printed my boarding pass and lent a pram for my baby.
The flat was comfortable, clean and baby friendly.
P.D. The bed was even better that mine ;)