Just one day. Arrive early at 6:05am on October 6th. Leave 24 hours later. For those that have been, what would you recommend for a first time visitor that really wants to get a taste of Iceland in a brief window of time?
I would visit any one of the city's geothermal pools (the Blue Lagoon is the most touristy and expensive, although not the only one) and spend time touring Reykjavik. There are lots of day trips to see the natural beauty of Iceland outside the city, but I don't think you'll have time to do it all (plus you may arrive totally tired and jet lagged). If you can handle a geothermal pool, that would be my top choice, followed by sightseeing in Reykjavik.