Birthday dessert!

To try some traditional Icelandic food, head to Lækjarbrekku (http://laekjarbrekka.is/).  It's well worth a visit. The restaurant is great and the service is as well - they may present you with a dessert like this one for your birthday. You can eat off of their preselected menus for a cheaper price (e.g. the lamb menu) if you are on a budget, and the food is just as good as the standard.

If you're NOT on a budget, and want to try out another fine dining establishment, the Sjavarkjallarinn ("Seafood Cellar") may be a good choice - It got its name because of its unusual location - in the oldest cellar in Reykjavik, but the style of cuisine is very "young" - taking basic Icelandic ingredients and presenting them in interesting, creative ways.