You haven’t said what she is studying, have you checked that there is no specialist software required for the course?
Last time I was studying (a year or two ago) we were recommended refworks which more or less required windows/word to be useful and while some people chose not to use it I definitely saved a lot of time using it to manage citations.
That being said take a look at ebay OS free – I’m rather tempted by the x240 but remember that their is no power supply. You could install Mint or have it run from a USB stick to try it.
Ebuyer has some Free DOS laptops but I’d not recommend a 15.4″ screen for university as its a but big – I’d recommend a separate screen and keyboard for home/dorm use
Have you considered a Chrome Book – just avoid the very cheep ones.
How about a iPad and smart keybaord (or Android equivalent)?
As a student she would qualify for an Apple educational discount – that may put a Mac Book Air in your price range (or take a look at the refurbished systems)