I would agree that a hinged end lid would probably work better than a side
hinged.  And don't forget the clearance issues to allow the lid to clear
the stone sitting up to be usable.

I've bought several stones with boxes, and almost all the vintage ones
appear to have been augered first.  One was nicely finished, and I can't
tell what was done first.

My one modern box is a nail together.

Kirk in HMB, who degreased a few tools this morning before my shower