Besides what valygrl recommended I noted the axe. There really aren't trees in Iceland to speak of (they cut them down years ago and almost collapsed the ecosystem). They are trying to replant so please don't cut any down.
I would recommend maybe an eye cover as it will not get very dark while you are there. Also ear plugs for all the wind. Bug spray or bug head net as the bugs up north will be bad this time of year. The midge flies will drive you nuts while you are trying to eat.
You are going to have an amazing time. My wife and I drove around most of the county and across the interior last summer. We were in the far north on the summer solstice. We were there for about two weeks and it never got dark the entire trip. Check out the second blog link in my signature for pictures and write ups from our trip. PM me if you have any specific questions about Iceland.