Yes Quicksilver, take lights. A small personal single cell light, maybe on a lanyard, which you keep on your person all the time should suffice for most tasks. Although it never gets proper night dark in Iceland during summer, the low sun during when it should be dark casts long shadows which can hide ruts, potholes and other trip hazards if you go out for a late stroll. I was there in late July/early August 6 years ago and managed to step right into a small icy stream hidden in shadows during a midnight walk, even though there was enough light to see the surrounding countryside.
A medium distance light with a balanced throw/spill beam would act as your main outdoor light. You'll be fighting against a certain amount of natural ambient light so 300 lumens wouldn't hurt, and dual mode would save battery power when full output isn't required. Whether or not you need a headlamp and a thrower depends on your intended activities, but going without any lights is pure folly.
As others have said, eye shades are a must (best advice I was given) otherwise it can be difficult to sleep, although I found myself getting up in the middle of the night just to witness the strangeness of it still being light outside. One thing I wish I had was bug juice. Because the summer is so short there, the insect life goes wild during its' reproductive cycle. Down in the south, the north and the non-glacial interior, you might be plagued by bazillions of midges. You quickly learn to keep your mouth shut. I was shooting video there and it really tested my nerve to hold the camera steady, tucked under my arm while feeling midges crawling all over me. I could see them in the reflection of my glasses, scuttling on my face and trying to get in my ears. I envied the stills photographer who could just shoot and run. When we were driven to some godforsaken airstrip in the middle of a barren volcanic plain, which turned out to be midge central, I noticed some seasoned tourists wearing mosquito net bags over their heads. An excellent idea.
Take a good camera because despite the bug life, the scenery is spectacular in the bright summer sun and almost cloudless brilliant blue skies. The melting snow means pretty much every mountain and hill will have waterfalls springing from all sides. If you can get to Gullfoss in the south west or Dettifoss in the north you'll be in for a treat. No midges either!
Even in summer, always pack your waterproofs in your rucksack. I forgot my over trousers on a cross-country hike back to the hotel after visiting a large volcano, and had to trudge back 8 miles in torrential rain which soaked me to the underwear and left me very cold, wet demoralised, and convinced that my blood was thinning by osmosis.
Booze is very expensive in Iceland so use your allocation of duty free at the airport, each member of your party bringing a different spirit so you can share and only need to buy mixers. Beer isn't really worth it but if you do buy a can it's fun to stick your hand over the top between sips and tell your fellow travellers it's so you don't lose any through evaporation - see how many take you seriously.
Of course you need to take a flashlight!! What will you do when you want to see under a desk, or in a dark closet, or in a windowless room when the power goes out? What if you get invited to explore a cave? What will you have to show off when you bump into a fellow flashaholic?? Sheesh! 
