Tie handwritten name tags to cookie cutters. You can also use the cookie cutters as napkin rings at the same time.
You can also use simple sugar-decorated cookies as "paper weights" for your place cards. Or press a whole into the cookies before baking them and tie name tags onto them once they're decorated.
If you have children at the table or you like things more playful, add some whimsy to the setting by getting some "little helpers" to hold your place cards / name tags.