Cafe Garðurinn serves the best vegan meals that I have eaten in Iceland so far. Every day, they have a different soup and main dish available. For 1300 ISK (about $11.50 in USD) you get 1/2 of a large tureen of soup, 2 slices of homemade bread (served with red pepper hummus) and a main dish. You can get a full portion for 2100 ISK, although I found the "half" portion to be perfectly adequate.
On the day that I went, the soup of the day was a Caribbean coconut red pepper soup and the main dish was spinach pie (served with cooked cabbage and sweet potatoes). Both were exquisite.
One caveat for vegans: Because the soup and main dish vary from day to day, sometimes one or the other may not be vegan. That said, the servers are very knowledgeable about the ingredients and, worst-case scenario, there are two other vegetarian restaurants within 1 block of Cafe Garðurinn.