Tour operators make a big deal about Grimsey, but the island on the Arctic Circle is something of a letdown -- a small, barren plot of land with a tiny village and a weird two-hole golf course. You can have much more fun in late June in Akureyri or the nearby towns of Husavik and Dalvik. All three towns are almost at the Arctic Circle, close enough so the sun never really goes down during the last weeks of June. They're all farther north than Nome, AK, to give you a perspective.
Akureyri is a pleasant city, the base for a slew of day tours, and the site of the Arctic Open golf tournament. It has a terrific public pool, some pretty good restaurants, and several small museums. Husavík, about an hour away, is Iceland's whale-watching center and the site of the bizarre Icelandic Phallological Museum. Dalvik, half an hour northeast of Akureyri, sits on a breathtaking fjord -- and is the departure point for the daily public ferry to Grimsey, if you still want to go.