I echo the sentiments here in that Iceland is a very safe place. Crime is very very rare, and even then tends to be of a petty nature.
There may not be a lot to see outside of Reykjavik in December, limited daylight (it is still pitch black at 10am) and "challenging" weather conditions will be a deterrent, but Reykjavik is a fun city, particularly for a young adult. There are lots of people your daughters age studying, travelling through etc and I´m sure she´ll have a great time.
I´m always scepital about the Northern Lights tours, in my experience if you go looking for them you won´t find them, but I guess if they´re not too expensive it could be a fun night out. I would definitely recommend that your daughter packs her swimsuit, there are lot of fantastic outdoor heated pools in Reykjavik and visiting them is a must. (much much cheaper than the Blue Lagoon).
I see people are recommending a trip to Akureyri. It is indeed a very pretty town but a LONG way from RVK.. (I´m thinking around 7 hours on the bus). Depending on how long your daughter is staying she may want to reconsider that. The flights will be crazily expensive.