Upvote:2
Overstaying can never be recommended as a good idea. Your daughter could be fined upon exiting the Schengen Area, but fines for minor immigration-related offences are often waived (if the foreigner leaves voluntarily).
Would going to London for a couple of days then coming back to Italy help?
It would, especially if she manages to leave the Schengen Area before her visa expires. If she only gets to the UK after a few days' overstay, it's probably better to book her a flight to the US directly from the UK, instead of her having to return back to the Schengen Area first.
What if she wants to visit Italy again to travel or study?
Yes, an overstay might negatively affect her future visits or visa applications, but probably not permanently. Since the US is a low-risk country in terms of illegal immigration to the Schengen Area, it's not likely she'll be permanently banned from entering it. IMHO (not an expert opinion), I'd wait at least a year before attempting to enter the Schengen Area again, either as a tourist or as a student.