I've updated this post because there is a whole Pinterest section with cookie exchange ideas. You'll find loads of tips on invitations, packaging, rules, menus, and recipe suggestions.
Here are some of my personal recommendations for cookie exchange cookies.
Oreo Truffles always go over well and are easy to make if you have a food processor.
Everyone expects and appreciates Easy Pecan Crescents (make with the best quality pecans you can find).
Texas Pecan Chewies freeze well and are fairly unique.
What would a cookie exchange be without Mexican Wedding Cakes? Make them with European style butter if you can find it.
Sachertorte Cookies are unique and fairly holiday-ish.