Pompeii is cool but that will be a very long day trip. I loved the Amalfi Coast, but I don't know if it would hold much appeal for 7 year olds. So I think I'd head north rather than south. Spend longer than you think you need in Rome - as you know there a lot to see, and I think having kids will slow the sight seeing down quite a bit (which is not a bad thing!). Then as another PP said maybe Florence or elsewhere in Tuscany, which would be a nice home base for day trips and just quieter/more relaxing. Then either Venice or CT, whichever you think your family would enjoy most.

Whatever you decide, it sounds like a wonderful family trip!

We did Rome - Florence - Cinque Terre. It's nice to build in something quieter, since Rome and Florence can be busy. The food in Bologna is amazing, Lucca is a pretty little town that makes a decent day trip/one-night. Florence means you can take a side trip to lots of little towns--Siena is pretty. I recommend Rome + Florence or Venice + small town.

Perugia is VERY kid friendly

In 12 days we did Rome, a small town in Tuscany (where we had a car for day trips), Cinque Terre, and Florence. It was lovely, we could have used another day in Rome and Florence but I loved the countryside. We were there in October, which was quite comfortable, but since you'll be there in the summer you may want to consider parts of northern Italy too. Wherever you decide to stay, be sure to ask about AC.

Venice would be at the top of my list! And, I would think a kid would really enjoy it, too. For one thing, you can just roam around and find something interesting to look at. You can "get lost" and have fun finding your way back. I've travelled many, many places in the world, and it is one of my all time favorites. I've been there several times and never been disappointed. I've also seen something different and new every time. I like the prior suggestion of Venice, Verona and the lakes.
Pisa would be fun for a kid, but I didn't know they couldn't climb under 8--so it might be disappointing.

Speaking only for myself, I would not go to Venice. But, I am much more of a home base person and travelling every couple of days to a new location isn't my thing. If you can, I second those recommendations of renting a place in the country side and doing day trips, with a few days in Rome at the beginning or end of your trip.

Semi-Crazy idea: You could stay in or around Bologna. Its about a hour to Florence, a couple hours to Venice, an hour to the Adriatic beaches.