What it teaches
Symmetry, which is what makes a butterfly a butterfly and a face a face. A child who has built one starts seeing it everywhere — and unlike most art ideas, it can be got exactly right or exactly wrong, which is what makes it a puzzle.
It is also spatial reasoning of the kind that shows up later in geometry: the fourth square from the fold on the left is the fourth square from the fold on the right, and working that out is real work.
What to look for
This is the game on the site that needs the least of a child: no reading, no dragging, no sound and no numbers. It works for a five-year-old and on the worst phone in the house.
Fold a real piece of paper in half and cut a shape out of the fold afterwards. That is the same idea in ten seconds, and it is the version a child remembers.