We had a wonderful stay at lucillas place. The apartment had everything we needed, including toys and books for kids. Lucilla collected us from the airport and left the basics in the fridge for one as we arrived late.
The Pigneto district has lots of good cafes and restaurants that were very cheap. There are also a number of small parks (one with little cars/trikes that our 2 year old absolutely loved). It is a bit away from the centre but is on the 81 bus to a lot of major sites. We found it easier to walk though with 2 small babies (buses and trams can be quite crowded in Rome). You can walk(at a brisk pace) to lateran basilica in 20 mins, Collosseum in 35 mins, trevi fountain in 50 mins (its as quick as taking the bus).
Only downside of the apartment for us was the difficulty in getting a double buggy in and out of the apartment. There are some steps at the entrance and the lift is very small. From staying in Rome before, small lifts are very common in apartment blocks and this wouldn't stop us from staying here again.