With respect Kev, it's a bit tricky offering buying advice with just a budget to go on.
That said, I've read a lot of good things about the video capabilities of the Panasonic Lumix GH series.
Specifically a used GH3 seems to fall into your budget, they are aimed more towards the serious video/vlog shooter, but are very capable with stills too.
It is a micro 4/3 system, shoots 1080p and has a tilt/flip screen....a must have for video....so I've heard
Here's a link to a decent used one at Harrison Cameras:
Used Panasonic GH3 - £325
Info page for the GH3 on Panasonic website
You'll need to factor in the lens cost too, which you should be able to do with what is left over....a decent standard kit lens would be a good place to start, but talking to your daughter about what she'll be shooting is probably advisable.
Here is an example of a standard zoom lens (MPB - £69) for M4/3 mount
(Links are for example only - more research is recommended)
If you want a brand new setup, then you'll be firmly in entry level gear, and all new entry level cameras will shoot decent video, broadly speaking.
You will be able to get a kit --camera+standard lens - for your budget.
Bear in mind, once you buy into a system (Canon, Nikon, M4/3 etc) you'll be sticking with it for a while.
Everyone will have their own views/preferences, and I'm not an expert, but hope this helps.
As Sphexx and Phil said though, she'll probably have a good grasp already of what she will need.
Good luck