Late summer or early spring are good times. The 24 hours of light in June and July is good, but it means that all the waking day is broad daylight. If you want to shoot for the beautiful light, it will mean shooting while everyone is asleep, all the restaurants and most of the shops are closed, particularly out in the country.
Yet in the spring and late summer, you still get tons of daylight along with more convenient shooting times. For example, in June and July, the good shooting times are from about 9pm until 8am or so...the rest of the time is full on broad daylight. Yet today, in late April, the sunrise is at 5:30am and sunset is at 9:30pm, so the "beautiful light" is in the early evening and early morning rather than from 12pm until 4am.
For example, this shot was taken around midnight on June 21st...ISO 100 film at F/4, so believe me, it is not dark! Slow shutter speed though...probably 1/15th or 1/30th:
That was right after the sun went down...it would be up again in 2 hours.
Anyway, the late spring and late summer area also good because there are fewer tourists. I would also recommend scheduling the trip during the week if possible...the Icelanders are big campers, and there can be traffic heading out of the city during the summer weekends.
One nice thing at least is that Iceland is only a 4.5-5 hour flight from JFK. It was quicker for me to fly from NY to Iceland that it was for me to fly from Santa Barbara to New York. Crazy, but true.