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If she's not the type to feel hard done by receiving a second-hand guitar for Christmas you'll definitely get more for your money and likely pick up something practically new anyway. I know when I was that age I'd have preferred a better used guitar than a cheaper new one!
I've been really impressed by some of the Epiphone guitars I've tried recently...
Epiphone Les Paul SL are comically cheap but actually seem really cool and it would free up the budget to allow for a nice amp to go with it! They do loads of colours too.
Fender Squier Deluxe Jazzmaster, just under £300 but will last her until she gets her first big advance from Sony... They sound great, cool colour, quirkier/more fun than a strat and a nice easy step sonically from acoustic as they do sound like proper electrified guitars... It's taken me thirty five years to get this...
Or spend a few extra quid and get her the J Mascis signature... The black AC4C1 is a great sounding budget amp option to go with it...
I realise I've totally blown your budget but do you really, REALLY love your child?