When I landed in the capital of Sweden, 1st of May 1994 it was -2 C° and a snowstorm. Snow in late may and june in Iceland as well as Scandinavia, is extremely rare, but not so rare in the highlands. It is about as the change that snow falls in London in April (which it did last April), as having snow in Reykjavik in May. I mean it can happen, everything can happen really. It is not the norm that snow is in may or june, otherwise it would be completely arctic and my elm, aspens, rowans and other trees in my garden wouldn't survive. To remind people who don't know it that southern and western part of Iceland have temperate mild oceanic climate. The temperature in Iceland has risen in last 10 years. Between 1930-1965 the weather was very mild, had good summers and relatively mild winters. During the early 1970's it changed. The summers got more cool and wetter and the winter got colder. This "iny tiny ice age" ended in 90's. This summer was record summer for temperature in many places. F.ex. the temp reached 26 C° in Reykjavik!!! The earlier record from agust 2004 was 24,5 C°. The temperature reached 29,8 C° in Þingvellir National Park. The average temperature in the southern part of Iceland during July (which is the warmest month) is around 11-13 C°. In northwestern, extreme northern pennisulas and eastern Iceland the average temp in july is around 8-12 C°, coldest on the extreme northernmost pennisulas and in the highlands or the interior of the country. The mildest part of Iceland is extreme south, and there it is rather rare to have snow, especially for prolonged periods. There, the grass can be green in march. The coldest month in Iceland is january but in some location, especially north, february is the coldest. The mean temp for Reykjavik in january which is the coldest month is around 0 C°, but in the extreme south the mean temperature in january is +2-3 C°, which is comparable with locations in northern England. The rainest month in October but the driest is May in some location with up to 260 sunhours. The spring is often very sunny. I remember April 2000, when the sun shined every day from 5th of April to the 25th of April. The wettest part of Iceland is near mountains of the Southeast with up to 4000mm a year but the driest is the Mývatn area with 250-350 mm a year.
If you don't like cold weather, end of June - early July is nice period to come. Then the nights are still 24 hours and the temperature is 12-20 C° or even higher sometimes.
