I like this particular Rush Hour implementation by Moebio: https://www.printables.com/model/109409-rush-hour-puzzle but needed something to store it in. 

It just slips in, has a small catch for resistance, and a spot for the printed manual or problems for reference.

The one shown is shrunk down for a 75mm board, but there's one I've made for the full size one as well. If there's a specific size you want, let me know since there are only a few parameters to change.

Print it as positioned in the STL. 

I used PETG for the catch to be flexible and not get worn out as easily, but I'm sure PLA will do as long as it's not too old or brittle.