Note 5 vs new budget phone for a kid?

I'd say a budget Samsung would be better. Go for a Galaxy A21S.

 

A good budget phone is fine. I don't think a kid would mind, except if he plays heavy games, maybe it would matter.

 

You can get a lot for around $200 - 300 these days in terms of decent specs. Does she need / miss the S-Pen. A pre - owned Note 10 Lite which only came out in January may be within reach, or a Moto G Stylus - I forget it's name in the US.

Without a pen, Nokia, Motorola, Samsung A21S as above, and Chinese phones are now very good indeed with excellent software.