Without hesitation, I must say The Blue Lagoon. I’m sure many people will back me up, but it’s such an industrial location that it’s quite mind blowing.
You’ll find that your skin feels fantastic for some time after too.

I think you’ll find it difficult to see the Northern Lights at that time of year or that would have been my second suggestion (though it snowed when I went and I have yet to see them!) with the constant day light, but I guarantee, you won’t be disappointed if you head for the Lagoon.

Hope that helps.

I absolutely second the Blue Lagoon. It was the best thing my wife and I did our first day in Iceland and may have been the best thing we did the whole time. I can’t wait to go back.

When you say a day and a night, are you just saying one full day? Or something more?

The Blue Lagoon is a must only because it’s basically paired with Keflavik airport, both being relatively far away from Reykjavik. I recommend going to the Lagoon before your departure if possible.. it’s a good way to relax into your flight. NOTE: it can still be near freezing and drizzly/flurrying even in June! However, the warm waters in cold air makes a nice complement to the typical summer swimming pool experience.

I’d bring an extra pair of swimming trunks and a small plastic or canvas bag to contain it. They will smell “egg-y” after your shower, since hot water in Iceland is geothermic, and so it has that sulfuric smell.

If you have one full day to spend, however, you definitely need to do a morning-to-evening tour of the Interior, hopefully visiting one of the desert glaciers. They pick you up at 8am where you’re staying in Reykjavik and bring you back by 7pm or so. The long days of June will give you the energy you need to enjoy it.