Hi Castellan,
I'm also a college student who loves shopping in Europe, but doesn't have a very big bank to break. :) I just got back from Rome and really enjoyed going to the street market by St. John Lateran. You can find all sorts of trendy clothing (a lot of it with a Boho vibe, if your daughter is into that), statement jewelry, knockoff bags/wallets, etc. for low prices, and you can always haggle with the vendors to lower the prices even further (especially if you are a young, pretty female!). If you start to walk away and pretend like you're not interested, they will pursue you and keep dropping their prices. I was able to snag a pair of aviator sunglasses for 3 euros and a collar-style necklace for 5 euros.
I also like shopping on the Via del Corso (near the Spanish Steps and Trevi Fountain) and the Via Ottaviano (by the Vatican). These streets have the typical, young-adult-centered chain stores that you see in every large European, American, and Canadian city, like H+M and Zara. I know chain stores get kind of a bad rap, but they do offer a more affordable version of European fashion. One chain store that's based in Italy that you won't find in North America is Calzedonia. In the summer this store is full of every color of bikini, and in the fall/winter it's full of uniquely-patterned tights. I picked up a whole bunch of cute printed tights at the Calzedonia in Salzburg, Austria during a semester abroad and got tons of compliments back home from friends, because they hadn't seen designs like that in the US. This store is all over the place in Rome, you can't throw a rock without hitting one! :) Calzedonia's sister store is Tezenis, which has more lingerie, nightwear, and workout wear.
Hope that helps! :) Have a wonderful trip!