Paper Selection
The paper can be anything you want. They really need to be somewhat heavy though. At least 65 lb weight to keep their form. Patterned paper would work OK too, but it would have to be double sided. Your imagination is your only limit.
Cutting
If you do is carefully you can even put text on these ornaments. Maybe make a simple 3D ornament for each member of your family with their name on it. The cuts are ornate so they are delicate. Be careful removing them from the mat.
I combined all the little pieces on one mat. To see how to do that watch my video.