The options are endless but since you are interested in the Cinque Terre, I would say spend Saturday through Tuesday nights in Rome. That gives you 3 full days in Rome & a nice introduction to Rome. (You will probably want to return another time to see more!) Wed morning, I would take the train to Orvieto & spend the day there (you will need to store your luggage somewhere). Orvieto is a great little town & it will be very relaxing after the hustle & bustle of Rome. In the late afternoon, take the train to the Cinque Terre & spend Wed-Friday nights there. I am assuming your flight out of Rome leaves early Sunday morning so Saturday afternoon you will need to take the train back to Rome and overnight there. If you can arrange to fly open jaw out of Milan instead of Rome, it would eliminate having to backtrack to Rome.
Rome - Orvieto - Cinque Terre will give you a varied itinerary (major historical city, hill town & Italian Riviera). The different pace of all these places might keep your 13-year old intrigued.
Alternatively, you could visit Rome, Orvieto & Florence. The CT is further than Florence so you will be spending several more hours in transit but I love the CT & can understand why you're interested in it. Florence is really great especially if anyone in your family is an art lover. If you visit Florence instead of the CT, you could take a day trip to Siena which is interesting too.
Either way, you will definitely leave Italy already planning your next trip back! Have a great trip.