Here's another attempt at creating a simple Origami toy puppet (well, it's not pure Origami if you cut the hair).
The design is based off a shallow box like:
You'll need a long piece of paper (around 3x1). Start by folding a long shallow box, but allow a bit extra paper on one end for hair.
Then just fold the box over on itself.
Draw a face, and you have a kids toy :-)