You are not going to get a "great" laptop for £300, but there are a few reasonable ones around for that price (or a bit lower). Try to go for a "name" brand like Toshiba, Sony, Lenovo, Acer etc.
There isn't that much difference in reliability of big-brand laptops...Latest Which? says that Acer is tops and Dell is at the bottom.
It's more important to match the spec (and therefore the capabilities) of the machine to the intended use.
£300 isn't going to get much of a laptop, unless you consider a nearly-new (refurbished) machine, when you could get a real bargain. You could try somewhere like this:
Their daughter gave them an old lappy. The OS was iffy. Could not reinstall as the BIOS was locked, so could not install new XP system. The admin password had long since been lost. Lots of advice on tinternet but it seems it is a lost cause, must have been 8 years old.