Why not Venice? It is magical. Okay, so no Venice, well then Cinque Terre would be my next pick and that is hard decision, because there is always Naples, Verona,Sienna, Palermo, Pisa, Lucca, Parma, Modena, Elba, Stromboli, Bologna, Positano, Capri, Ustica, Milano, etc., etc,.. As a 19 year-old I think Verona would be interesting. Not only is it an amazingly beautiful city, but hello, Romeo and Juliet slept here. You can't miss with the Amalfi Coast either. Wow..Good Question...still I have to go with Cinque Terre...
Lakes region or Dolomites are options.
It really depends on how many days you have to spare whether you can visit another place. Have you been to Venice? If not, its not to be missed. I was put off by it because of the cost and the overcrowding, but I finally went last summer and I wish I had stayed more than just 2 days. It is pricier than other cities but there is no other place like it!!! If you venture away from the main sites it's not terribly crowded .
What does your daughter like to do? Beaches, hiking, biking, shopping, cooking...?
My vote would be for Lake Como, especially a stop in Bellagio. We did some shopping there, then had lunch by the lake. Breathtaking!
Yes, it's important to know your interests, and also how many extra days you have for this third place. Do you want a place off the beaten track, or are you willing to beat it a bit more?
There is soooo much to see and do in Italy. It's a truely breathtaking country as the other posts have pointed out. I could spend months and months touring Italy and never get bored!!
There is a video called "Visions of Italy" that I highly recommend. It's a wonderful way to get ideas on where to go and a great way to remind you of your trip after you return. You can purchase on Amazon.com.