What's the good reason for supporting Barry? His opponents. The three other Democrats in the race are D.C. council members, all of them genial, mildly competent, utterly forgettable. The main Republican in the race, Carol Schwartz, is also a genial mediocrity. Schwartz has made her career by losing to Barry--the Washington General to Barry's Harlem Globetrotter.
The field is so weak because District politics is a dead end. It's a small city. Congress exercises veto power over everything the mayor does. There is no higher office to graduate to. The mayor can't rise to the Senate or the governor's mansion, because the District has no senators or governor. It is full of smart, ambitious folks who might make wonderful mayors (Deputy Attorney General Eric Holder, to name one), but they ignore local politics in favor of federal jobs with real power. Barry is the rare talent who stayed in local politics.
Forced to choose between Barry and his uninspiring rivals, is it any surprise that thousands of Washingtonians picked the mayor? He may be a scoundrel, but he's still the smartest politician the city has.