I studied abroad a few years ago. Honestly money would be my main suggestion. Or a nice camera. The impulse to buy things she needs and send them is high. Resist this impulse. Especially with things that will have to be taken and brought back. Shipping and checked luggage gets expensive fast and dragging it around gets old.
Only send things she uses frequently at home and will need before she can buy them. The cost after paying to get it there and the pain and suffering of dragging it through the UK first wont be worth it even if the item is more expensive abroad.
If I did it again I would only take my most favorite things and things that would be super useful/too expensive and buy everything else there. For my clothes I would take maybe 1/4 of what I originally did and buy what I needed/wanted there. I wanted to buy things anyway but felt bad because I spent so much money getting my clothes to Europe. If she is picky about the fit of something (especially jeans) take those, but otherwise be really selective. Plus these things make awesome souvenirs.
A small medicine pack would be great. You can buy what you need in Europe, but its nice to have it initially. One note (and this may be tmi, but its important) if your daughter is prone to UTI or yeast infections send her with OTC medication, my friend got one in Italy (Florence, actually) and the Italian pharmacist thought she was asking for yeast rather than yeast infection medication. He gave her some weird yeast cream/paste/product thing. She was very lucky she translated the package before use.
I also recommend a
small package with her most favorite school supplies and candy. I really like a particular type of bic mechanical pencil. I thought I would obviously be able to find something similar in the UK so I brought maybe 5. I could not find anything remotely similar. I hoarded them like gold until my mom sent me extra in a surprise package.
Don't by a hairdryer here and take it. It will die quickly. They are one tool that can't take the power conversion. I recommend buying in-expensive styling tools and then seeing if she can donate them to charity/a friend/another student before she leaves.
A good idea if you have the money (and it will cost $$$ as international shipping is outrageous) could be an IOU for a package of her most favorite things that she misses from home. That t-shirt she forgot, a candy, a specific type of potato chip or salad dressing. Collect ideas/wishes when you talk to her and mail it a month or two into the program. Ship it in an extra duffle bag and bam - extra bag to bring things home in (if she remotely likes things/shopping she will likely need this anyway). You/she should also be able order things from the Italian
amazon website (I used amazon.uk a few times, quite lovely) if she cant find them locally.
I took one large suitcase and my carry on bags. For the backpack I took my regular Loew Alpine that I used everyday for school. I'm not sure which ones you are looking at, but the large backpacking packs will need to be checked. If she isn't planning on doing a lot of traveling I would recommend a regular suitcase. If she takes the pack, be aware they shout "I'm a tourist! I probably have money and valuables" even more than having a large suitcase. I have one which I have rarely used (also Lowe Alpine), I traveled around Europe for three weeks and took a medium/small rolling duffel and my backpack. They can hold a good bit depending on the size you get but less than a regular suite case (in my opinion), I really recommend going into a store and looking at them since 36L or 55L don't give you a great idea on how big the bag is and they come in all sizes. Plus if your daughter is small a large pack may be too much for her to carry. I think a regular backpack in around 20-25L? I've also heard great things about LL Bean, North Face, and Patagonia backpacks.
If she likes reading on the ipad then digital travel books are the way to go. Physical books take up so much room in your luggage and are so heavy. I couldn't bring all mine when I traveled about while studying abroad and found that the ones I needed were the ones I didn't have. I love Rick Steves books and ones with good pictures (eyewitness I think?).
I really love Columbia Rain Jackets. They range from about $40 and up for a shell, and keep you very warm with a fleece jacket underneath. Some are even layered to be warm by themselves. I have a green Columbia I got around 2008 and it needs to be replaces soon, but I was happy with it. For a bit pricer (ie around $100+) I actually took a black Patagonia with me to Scotland, I have abused this jacket and it has held up well (it also needs to be replaced or re-waterproofed). Shell's are more lightweight and they are just the waterproof outer coating without an inner lining. If she doesn't have one I think it would be a nice gift.
One thing I recommend not getting in the Eurail pass. It can be a complete waste of money especially when traveling on local trains (cheaper tickets). I came out about even on my pass, but the added stress on not losing the stupid thing/making sure I didn't violate any of the rules wasn't worth it. Plus in Italy you still have (or had) to pay to reserve the train seat even if you have the pass. If you don't they can fine you. Luckily a conductor took pity on me when I was traveling from Florence to Venice and only made me pay the reservation fee.