I'd especially want to know the age of the kids and the budget. I have had two good meals at Ristorante Riflessi, http://www.ristoranteriflessi.com/, but it's somewhat expensive and probably too sedate for a family with very small children. I also enjoyed Ristorante Accademia, http://www.ristoranteaccademia.it/, and plan to return there later this month. Might handle a family better than Riflessi, but still relatively expensive.

On the other hand, for a family on a budget with several small children, I'd consider some of the small trattorias along the Piazza dell'Independenza--I wish I could remember the name of the place I went, but I basically just stepped in to several places for a moment and stayed where I found the most folks, including wait staff, speaking Italian. I made my very broken Italian work. The food may not have been amazing, but it was very good, and it was very reasonably priced.

Perche hasn't mentioned it in this thread but has in others that there are many reasons to take restaurant reviews in Italy with much more of a grain of salt than in other places. I think that's valuable advice to consider.