Dave, I am no expert but if singing is something that she really enjoys, a case could be made for the guitar as it is very easy to learn to play a few chords to accompany youself. I suppose some of that would depend on what she likes to sing but within folk music there must be thousands of songs that can be accompanied with 3 chords.

I am no expert on music and I'm not a parent but I think that if I was in what I think is your situation, I would be thinking in terms of a small nylon strung classical guitar and it hould be possible to find one that sounds nice and plays well fairly cheaply. UK wise, I have seen somthing along the lines I am thinking of for around £40 new.

Jon