I rented a 172 in Reykjavik a few years back, and it was an excellent experience - expensive, but excellent! I took my wife and daughter, and one of their instructors, who went for the ride free of charge, to visit his buddies where I was going anyway (that was easier than the checkout process, though that was offered to me). Take cash to pay for the plane. I honestly cannot remember the flying school name, but it of helps you to search, the plane I flew was TF-FHO.
From Reykjavik, fly south east a bit over an hour to Vestmannaeyjar, Walk to town from the airport, absolutely beautiful, and you can hike on the volcano of 1973. From there, you can loop north over the icecap, and back to Reykjavik. You'll have seen nearly a quarter of the island if the vis is good, and about three hours flying.
If you need more info, you can PM me, and I'll inquire of an Icelandic friend of mine