Here are some random unorganized thoughts from my trip in 2018.
No northern lights in June as the sun never really sets. That said, since it is daylight almost the whole day, you can do many hikes at off peak times in full light while avoiding the crowds. I was going to say sunlight, but it was pretty gray even in June. Here is a picture I took close to midnight near Reykjavik .
Go to the Blue Lagoon when you arrive if you have arrive in the morning. It's a great way to refresh. The sit-down restaurant was quite good.
Bring lots of cold weather clothing, especially rain gear. Weather is very iffy. I don't think it got over 65 when we were there.
Don't skip a gas station if your tank is 1/2 empty. I realized that when looking at my KM to empty and realized that the 2 nearest gas stations were within 10 km of empty. That was never wracking.
I am not a bird watcher, but I would go out of my way to see Puffins.
If you get a ratty looking rental car, embrace it. I was disappointed to get an old Dacia Duster with 117k km, some dents, cracked windshield from Avis, but by the time I dropped it off, I was super thankful that the car had been previously abused by someone else. Here is was the day before I dropped it off. We washed it to prevent issues.
I am hoping to go back soon. Book it!