My votes would go to Yamaha, Cort and Ibanez for quality and consistency. S/H their more basic models will be not too far beyond budget. I've no personal experience of Hartley Benton instruments but notice that they often review well.
Set-up and reasonable quality strings can elevate a cheaper bass enormously but, unless you do it yourself you'll be adding a significant bill to your budget (£50-£80 including strings). This is lost money. Therfore, imo you'd be better off getting a S/H instrument that doesn't necessarily need a set-up and strings and would retain the majority of its value (try first if at all possible); if I had to hedge my bets I'd go for Yamaha, having never owned or played a bad one.
All the best to your daughter as she embarks on this exciting but dangerous quest!