REYKJAVIK—Jon Gnarr, leader of Iceland's newest and most offbeat political party, promises various changes if he is elected mayor of Reykjavik: a polar bear for the city's petting zoo; palm trees for its icy waterfront; free towels at its swimming pools; a rearrangement of statues; and a commitment to "sustainable transparency."

These aren't exactly pressing issues for a city reeling from one of the biggest financial meltdowns in history. And yet, according to polls that have...