Akuyeri is a cool little town

Forest Lagoon is an awesome geothermal spa. They also have an outdoor geothermal swimming pool with waterslides and such (open year round). Plus, you can catch a ferry up to Grimsey, the only part of Iceland that lies within the Arctic Circle (three hours each way, and it can be a rough ride, but it'll allow you to say you've been inside the Arctic Circle). Akuyeri is a 5-hour drive from Reykjavik, though, so it's a commitment.

You could drive south and check out the black sand beach at Vik. There are a variety of other outdoor sights in the area as well. Vik is about three hours from Reykjavik.

I was lucky enough to live in Iceland for a year, so I got a chance to explore the whole island. I get that these ideas might not make sense of you're only spending 4-5 days in country.