Steven,
While there are never any guarantees, in all likelihood your daughter will be fine travelling on her own. Many other young women have done the same thing without any problems.
However, a few points to mention. A previous reply mentioned the Guidebook Europe Through The Back Door. I think that should be an essential pre-read before she goes to Europe. Pay special attention to the Rail Skills chapter, and I believe there's a section devoted to "Women travelling solo" as well. Use of a Money Belt would be a really good idea, but be sure to tell her to wear it properly under the clothing, unlike several young tourists from the U.S. that I observed last year in Italy who were wearing their Belts externally! I was going to say something to them, but figured it would like "talking to a brick wall", so didn't bother.
You might find that having her travel with at least a basic Cell Phone would provide some "peace of mind" for you and your wife. There are lots of posts here on the topic but if you need further details, post another note. I always travel with a Cell phone and sending brief texts to family back home is relatively inexpensive (even when using roaming).
If her travels after leaving the group are well planned, that will help to minimize any "unexpected problems".
Staying in HI Hostels should be quite safe. As they're part of a chain and have to meet certain standards, there's more structure and therefore they're not "party Hostels".
As a parent myself, I can certainly appreciate your concerns. I was a bit worried when two of my Sons went overseas (including one who went to the U.K. at 18-years old); it's often a "leap of faith" letting children explore the world on their own.
Good luck!