These are old fashioned, and a bit off the norm! My family and friends LOVED them!

http://www.gourmet.com/recipes/1990s/1996/09/anise-scented-fig-and-date-swirls

The dave Lebovitz Nori shortbreads are also pretty good.

Oh I LOVE those date swirls too. They are pretty much my favorite cookie.

These crunchy sesame cookies are REALLY good, and easy to make. It uses Tahini.

http://find.myrecipes.com/recipes/recipefinder.dyn?action=displayRecipe&recipe_id=1687690

I dunno if these count as cookies exactly, but the Samoas bars that were featured on this site a while back got a lot of "oohs and ahhs" when I made them.
http://www.thekitchn.com/thekitchn/inspiration/weekend-treat-samoas-bars-082289

Not terribly odd, but a nice twist on a standard chocolate cookie recipe... and yummy. Double Chocolate Fig Cookies:
http://www.eons.com/groups/topic/408786-Double-Chocolate-Fig-Cookies

I make chocolate walnut cookies that are baked inside 3-dimensional molds that look like half a walnut shell. The cookies are then "glued" together with chocolate buttercream. People are usually pretty impressed with those!

As well, there are the Swiss cinnamon stars (no flour -- nuts instead), linzer cookies made with very pretty cookie cutters and raspberry jam, vanilla crescents, "checkerboard" cookies... Most of these recipes are old family recipes or from cookbooks (e.g. the checkerboard ones come from Nick Malgieri's cookie book, and the Swiss cookie recipes from an article he did in Gourmet in the 1980s).

My favorite cookies, aside from the aforementioned oatmeal raisin, are these molasses cookies from here at the Kitchn:
http://www.thekitchn.com/thekitchn/dessert/recipe-mini-molasses-cookies-with-lemon-filling-017251

This Gourmet article listing their favorite cookies should also provide some good ideas:
http://www.gourmet.com/recipes/cookies

Engifer Smákokur (Icelandic Ginger Cookies)

recipe on my flickr page. The photo leaves much to be desired, but the cookies sure are good!

I also recommend checking out The French Cookie Book -- the best book on the art of cookie making, as only the French can do. I cherish my copy!

http://www.amazon.com/French-Cookie-Book-Bruce-Healy/dp/0688088333

Here are Nick's checkerboard cookies, just as I make them! (the dough is particularly delicious, and lends itself to other types of cookies as well):

http://thebarmybaker.blogspot.com/2006/12/going-boldly-where-no-barmy-baker-has.html

I use a recipe that's similar to oatmeal raisin but instead of raisins it calls for dried cherries (though I have used raisins anyway) and cardamom. I found the recipe here: http://www.amateurgourmet.com/2009/02/baked_oatmeal_c.html

And I make wonderful cake-y sugar cookies like my grandma made, but that one's a family secret. I haven't seen it out there yet. ;)