Fuji HS10.
It is a great all round camera, one that she can do pretty much everything with, without having to buy another lens, which makes it a great starter camera.
Ok, she will lose a little Image Quality in some situations, but what is going to get her intersterested, only having 50mm -200mm as a max range, no video, no macro no live view?
With the HS10, she has a massive zoom - 700mm (ok, handheld at the long end may be a struggle, but how often you gonna use 700mm), a very good macro range, full manual mode, will take a flashgun, will shoot RAW as well as Jpeg, uses AA batts, takes filters etc etc. It also shoot video, and also, unlike the other cameras mantioned has a proper twisty zoom.
It even has a flip out screen.
If she has this camera, it looks and feels like a DSLR, so will make her feel like she has a proper camera, rather than a compact. She will get better shots than her mates on their phones, and with a £350 budget, Arefurb from the fujishop at £240 leaves enough so she should be able to get the camera, memory, rechargeables, a cheap tripod, bag etc etc.
It is worth considering this, although the DSLR brigade will say you may as well just get her a basic DSLR, think how much more impressed she will be when her camera has most of the features of a top end dslr, so much more likely to take up the noble art of togging!!