I bought a new mobile a few weeks ago. My pseudo-scientific method for choosing one was to go to the Argos website, place all of their SIM-free phones into ascending order of price and then to seek out independent reviews for each one online (as well as reading the reviews on the Argos site), looking for the first one I came to that had got really good reviews. All of the reviews I found were pretty mediocre as the prices gradually increased until I suddenly stumbled across one phone that was rated far, far better than anything I'd looked at up until then. I paid £169.99 for it, which is well within your budget, but it's come down by £20 since then, to just £149.99. I've been using it now for a few weeks and I'm absolutely delighted with it. So that's the phone that I'm going to recommend to you now:
https://www.argos.co.uk/product/8886635
As it's SIM-free, you can opt to use any PAYG provider. I use Asda Mobile, where I get a whopping 10 GB of data each month, plus unlimited calls and texts to UK numbers, for just £10. See here:
https://mobile.asda.com/bundles
(Click on 'Standard Bundles' to view the full range of cheaper options. Note that those are PAYG 'bundles', not contract deals, so there's no contract involved).
If you do decide to go with my recommendation, buy a sturdy case to go with it. 11-year-olds can be pretty rough on phones! Here's what I bought, which is an excellent product and fantastic value too:
Amazon.co.uk User Recommendation