Iceland Stopover

I would advise you to spend some time (I would say a majority of your layover) in the city regardless of where you choose to sleep for the night (either in Reykjavik or the airport area). There is nothing to do around the airport area, so keep that in mind, whereas Reykjavik has tons of nightlife, dining, and sightseeing options (and you can walk at any time of day and night - it feels very safe). I would definitely not miss seeing the city on your stopover - it's a really neat place.

Here is the link for the bus to take you to the airport (either in the evening or the next morning or just to the Blue Lagoon). The Flybus is coordinated with all flights, even those early in the morning.
https://www.re.is/flybus/
https://www.re.is/day-tours/airport-to-blue-lagoon-no-entry

Don't stay in the airport area. Go to Reykjavik. Between Reykjavik and Keflavik (airport) there is Blue Lagoon, little closer to the airport than to Reykjavik. There is a modern hotel at Blue Lagoon you can spend overnight there. The hotel is somewhat expensive but the entrance to Blue Lagoon is included in the price so it comes about the same as cheaper hotel plus entrance to Blue Lagoon. Reykjavik has feeling of a small town, very clean with the clearest air out of all capitals I ever been to.

If hitting the Blue Lagoon first thing was a priority, you could even arrange to go there direct from Keflavik Airport, then head to Reykjavik after your Lagoon visit.

If going directly to/from the airport and Reykjavik, the Flybus is definitely the way to go, and will drop you off/pick you up at whatever hotel you're staying at.

We actually went to the Fontana Baths & Spa during our visit in April 2015, and didn't go to the Blue Lagoon. We stayed at the Icelandair Hotel Marina, right by the Reykjavik harbor sights and activities, and highly recommend it. Also, if How To Be Icelandic in 60 Minutes is being performed at the Harpa Hall, when you're in town, consider going to the show - very funny and educational, too! ;-)

Thanks everyone! This is great information. I will go ahead and keep my reservation at the Captain Reykjavik Guest house. It was rated well on Tripadvisor. I will have to go there first as my plane gets in too early to go to the Blue Lagoon. I will look at the other bath house and the show. I am excited to see Reykjavik now!

Reykjavik is definitely worth it. It would be easiest to go the Blue Lagoon first, and then head to town. Otherwise you'd have to backtrack.

Plan some time to see the city - there are interesting sights. I took the Grand Circle evening tour. We left Reykjavik around 8 p.m. (it was June) and there were few other tourists at any of the stops.