I found out how to make these woven paper balls on the Cheese Thief Blog.
Go there now! There is a fantastic video tutorial.
These could be made from old packaging, sheet music, wrapping paper... I'm even thinking they could be made from fabric - if I stuck fabric to paper and then cut it? That may require Vliesofix to fix the fabric to the paper - just like in this tutorial on how to make fabric cards from Frog Goose and Bear.