Do be aware that Uber has had its fair share of safety scandals, including drivers implicated for rape. In these cases, Uber's response has been less than stellar, with a lot of media back-lash.
Haven't used them myself, but I also know they recently got into a lot of trouble with the City of Cape Town recently, with a lot of Uber taxis impounded (because they're not properly licensed as operators).
Uber is more the technology platform, not a taxi company. In my opinion you're still better off going with a reputable taxi company, but you might end up getting a cheaper deal with Uber (although did you hear about that kid in Cape Town who paid like a serious amount of money over New Years?).
Not to scare you or anything, but if it was my daughter I might end up rather phoning the most expensive taxi company I could and paying for it happily.
Edit: Link to Uber story -
http://time.com/3626143/uber-india-rape/