Jónsmessa believed to be June 24th
In the old days, it was believed that during one night every year the Cows could speak a man language. It was on Jónsmessan (Midsummer night), which vas believed to be June 24th. It is a special day in Iceland. The story sais that you should also roll around naked in the grass during the night and make a wish and all your wishes would come true. It is also believed to have a healing power to do this (only in Iceland).
June the 21st however was the longest day of the year as it was daylight 24hours. That is one of the good things in Iceland, the dailight is longer and you can explore more each day.
There we have at least two good reasons to travel to Iceland in June.