One present my daughter received for Christmas was this animal origami set (purchased from The Land Of Nod). I keep the set in our church "quiet" bag. The set comes with stickers to decorate the faces, but it is just as fun and easy to make one's own from scrap paper and draw faces, spots and details, which we'll do once we run out.
I chose the two most basic animals to share. You'll have to excuse the photos, they are a bit blurry. Before starting, make sure your paper is cut into 8 inch x 8 inch squares. The purchased papers come with dotted lines to indicate where each fold should be. It would be simple and maybe helpful to draw a light pencil line for a child.
Fold corner to corner to make a triangle
Fold one corner down to make an ear
Fold the other corner down to make an ear
Fold up the point at the bottom, behind the face.
Fold corner to corner to make the triangle
Fold the corner on the left in and the corner on the right in to make creases.
Then fold them behind the face.
Fold up the point at the bottom.
Lastly, fold down the very tip of that triangle to create the mouth/nose.
My daughter's are still learning how to fold the animals, but they do enjoy watching me do it slowly and then decorating. It makes a great quiet time activity.