Here is a short tutorial on how to fold a snowflake with scratch paper:
1. Starting here with an 81/2" x 11" piece of printer paper.
2. Fold the top corner over to the side to make a triangle.
3. Cut off the extra at the bottom.
4. Fold both corners up to the point of the triangle.
4. Fold in half, and trim the bottom edge to make it round (if you don't you will have a square or diamond-shaped snowflake). You can also use a hole-punch, or pinking scissors to make interesting patterns.
Origami is fun and also can be a good brain exercise. We have several origami books we've gotten ideas out of, but here are a few online resources I can share with you for a Christmas tree, a stocking, a star, a Santa, (and different Santa here - this is more like the pattern we have in our books).
Origami paper can be expensive, but you can make your own by cutting a piece of scratch printer paper as I described in the first step for making a snowflake. Color one side with crayon to make it a different color.
Wycinanki and Paper Garlands -
Another fun thing to do with scratch paper is Wycinanki, or traditional Polish paper cutting art. There are many ways to do it, but an easy way to start is to fold a piece of paper in half, and lightly draw out a scene. Make sure whatever is draw on the center part is half of what you want (so you don't end up with a two-headed Christmas tree or something). Cut out around the lines and unfold it. You can then glue it on a darker sheet of paper, or hang it up on the window like a snowflake. When you get the hang of it, you can layer different colors of paper together. They can be very complex and decorative.
Garlands of little Christmas trees, angels, stars, or gingerbread people are easy to make by folding a piece of paper accordion-style, and drawing out whatever it is you'd like to cut in light pencil on the front. Just make sure both sides stay connected at some point, or you will end up with a bunch of little gingerbread men or angels running around loose and not holding hands together.
In the past, I have had very little success in interesting my children to sit down for snowflakes or other paper crafts if I get everything ready and announce that we're going to do such-and-such. They usually aren't interested. The best way I have come up with is to get everything set up nicely, and sit down and start working on it myself in an absorbed, I-hope-no-one-will-bother-me way. They will come running in just a few minutes!