There are a hell of a lot of options and you need to decide on the form factor you want. Have a look at this video for a general overview of the possibilities at every budget. Whatever you decide on, I would recommend getting it on eBay to avoid delivery charges, tax, etc. You'll get a lot more value out of used "like-new" products.
You could use an old Android phone and this might be a most versatile option....although locking down might be an issue (depending on how young this sibling is).
For emulation consoles:
Miyoo mini is probably the cheapest option in your budget for you that can play what you want. Also look at the Anbernic 35XX device range (same hardware in a ton of formfactor options).
I would personally look for a deal on a Powkiddy X55 (or TrimUI Smart Pro is a bit cheaper) to get a candybar form factor and up to PSP games. Retroid pocket 3 pro might be a good option too if you can find a bargain.