Puzzle/game of NP class for a high school student
There is a list of a bunch of games that are NP-hard here: https://www.ics.uci.edu/~eppstein/cgt/hard.html  You might also enjoy this list of Nintendo games that are NP-hard: https://jeremykun.com/2012/03/22/nintendo-np-hard/.

Some well-known games that can be proven to be NP-hard (in some form or other) include Tetris, Minesweeper, Checkers, Chess, Dots and boxes.

I think Dots and Boxes might be a particularly surprising and accessible one -- it's rather mind-blowing that such a simple-seeming game hides such complexity.

Caveat: do understand that in many of these cases, we're actually considering a variant of the game.  Many games are finite, so there are only finitely many positions, and thus can't literally be NP-complete -- so instead we consider variants.  For instance, instead of chess on a 8x8 board we consider a variant with a $n \times n$ board, or something like that.  The general point still applies but there are technical details that you should make sure you understand before using this in a classroom.