Al 34 is a pleasant and bustling little place near the Spanish Steps (Via Mario de' Fiori, 34), frequented by locals and vistors alike, and very welcoming to children. TripAdvisor reviews here.

Etablì is a cafè/restaurant/wine and cocktail bar located at Vicolo delle Vacche 9/A.  TripAdvisor reviews here.

Pizzaria Ciro is a good bet for reasonably-priced family-friendly dining near the Spanish Steps. It's at Via della Mercede, 43. TripAdvisor reviews here.