When Representative Vito J. Fossella strode into a Staten Island senior center on a recent snowy afternoon, he was greeted as something of a conquering hero. After entering the dining area, he was introduced by the center's director, who mentioned that he was a possible candidate for mayor. "Maybe he'll run for mayor and we can get rid of Bloomberg," the director said. With that, the group of about 50 people erupted with wild cheers and sustained applause. It is not uncommon these days for Mr. Fossella, a Republican from Staten Island, to be described as a possible challenger to...