If you are party oriented then you don't have to feel guilty about sleeping late because the beaches don't get crowded until mid-afternoon and you can still work on that badge of health, your tan, until after 8 pm.
After a nap and dinner, club life begins after midnight. Although in high summer you'll find bar or clubs on most any Greek Island with over 500 people living on it year round, islands that are particularly well known as places to meet other people, dance, drink, lay around on nice beaches are: Mykonos, Paros, Ios, Santorini, Rhodes, Crete and several more islands described here.
On the other hand those of you with a different approach to activity will no doubt want to get up with the 5:30 sun to enjoy the moderate heat of the early morning while you get that long walk or jog under your belt. The heat of the day is best passed in a prone and shaded position. Hikers and others may want to consider islands that receive fewer tourists. Or traveling off season.