If you are arriving early in the morning in Iceland, you might consider doing a tour of the Reykjanes peninsula. This is the area surrounding the airport. There are a few infrequent group tours that might fit your schedule on the way to or from the airport. They will pick you up or drop you off at the airport and/or Reykjavik. We did a private tour which was about $300-400 but worth it. They picked us up directly from the airport, we went to VikingWorld for a bite to eat and a quick look around the museum and then spent about 4 hours touring the peninsula, stopped briefly at the Blue Lagoon to check it out, and then dropped us off at our hotel in Reykjavik. I really enjoyed the stark beauty of the Reykjanes peninsula.
If you prefer, there are also some express Golden Circle tours which only take 6 hours from Reykjavik. I did an evening one in July which was very nice as the crowds were fewer and we had a smaller tour group.
In Reykjavik, I really enjoyed the tour of the Harpa concert hall. If you like history, both the Settlement Museum and National Museum are worth visiting.