I believe that literature is the real Icelandic heritage. That is the one thing that has been preserved throughout the ages, unlike most of the stuff the tourist shops try to pass off as traditionally Icelandic. About 10% of Icelanders have published books (myself included, in fact). Icelandic literature is a source of great pride for us.
So here are a few recommendations for books that you might enjoy. The list includes great examples of classic Icelandic literature and some modern stuff as well. These should be available in any decent bookstore in Iceland.