Arch linux? :) joke aside perhaps something with btrfs support is handy. you can easily rollback if something breaks. For parental control don’t give the kid sudo/root. other then that restricting websites and stuff is more easily done on a firewall outside of the kid its control.
Almost any of the larger distros will suffice i think. Personally a fan of opensuse tumbleweed which has btrfs support out of the box. use a DE like kde/gnome and i think you have a very solid start