We only had a day unfortunately, but I’d definitely do the little day hike from Arnarstapi to Hellnar along the coast. Rauðfeldsgjá is also a cool little stop, a very narrow ravine. Depending on how into rock climbing/tight spaces you’re in you can probably sneak back into the ravine quite far, but I didn’t have good shoes on for dealing with going up the stream too far and I had my camera I didn’t want to risk getting wet.
Both places are sort of heavily trafficked unfortunately, but if your background is dealing with national parks in the western US it really wasn’t all that bad. Apparently when we arrived about 2 weeks ago, peak tourist season hadn’t quite hit though, so maybe there’s been a noticeable increase in just that time (since the window is pretty small it seems).
We stayed in Grundarfjörður, which was near some solitary hill that is in Game of Thrones (according to my wife, who read something about that but hasn’t seen the show). I’ve seen the show and have no recollection of it, but then, I have no recollection of most of the show. If you’re into that, it might be neat though. It was cool to me regardless. There was a nice waterfall nearby too: Kirkjufellsfoss. We lived it up food-wise on the trip and ate at some insane restaurants and while Bjargarsteinn wasn’t the BEST we had there, it was really good. Plus, it was right in the harbor and with a view of Kirkjufell from most seats.