We went in July a few years ago - loved it. It was just my husband and me, no kids. That time of year, the weather is as warm as it gets (50s-60s) and it's light about 22 hours a day. Here is an email of stuff to do I wrote up for a friend who was visiting after me:
Blue Lagoon – A thermal pool-turned-spa created by the crossover of some natural elements and a nearby power plant, it's supposed to have healing powers (dubious, but it was fun). This is out by the airport so it’s a good thing to do on the first or last day in town (we did it the first day since we were a little jet-lagged anyway). We used Reykjavik Excursions for our transportation (airport to Blue Lagoon to downtown Reykjavik), the package also included entry fees for the Blue Lagoon and got us from the airport to downtown, so that worked well for us.
Golden Circle Tour – There are several of these that hit some of the natural wonders in the area outside of Reykjavik, including a huge waterfall (Gulfoss), the town of Geysir (named for a geyser that goes off every ~5 min), and Pingvellir national park (which is a historic site in the founding of Iceland as well as a place of geographic importance where you can actually see the divide between tectonic plates). Several companies offer tours of these sites (we used Reykjavik Excursions again) or you could probably drive it on your own. Takes the better part of a day to do the drive and see the sights.
We used Arctic Adventures for a hike to a hot spring, they had several adventures and excursions.
There is also a lot to do on the water near the harbor in Reykjavik – puffin watching, whale watching, etc. Even if you don’t take a boat into the harbor, it’s neat to go down there. There’s also an interesting maritime museum near the harbor.
Downtown Reykjavik is a cute little town – very walkable and lots of shops and restaurants. I’d recommend walking around the large lake in the middle of town and going inside the big cathedral (Hallgrimskirkja).
A couple of restaurants we liked are Geysir (
http://www.geysirbistrobar.is/) and Grillmarket (
http://www.grillmarkadurinn.is/) – both are right near the city center (*not sure if these are still around).
We stayed at the Radisson Blu 1919 and it was perfectly good and very close to the city center – but there are tons of hotels/hostels and the whole city is really hospitable and welcoming of tourists.
Have fun!