The online video that launched former U.S. Rep Anthony Weiner's New York City mayoral campaign Wednesday featured someone who until now had played a limited public role in his political life: his wife, Huma Abedin. Ms. Abedin opens the video in a shot that shows her and Mr. Weiner feeding breakfast to their son, and she closes it sitting beside the candidate on the stoop of a brownstone. "We love this city, and no one will work harder to make it better than Anthony," Ms. Abedin says to the camera. As a longtime aide to former U.S. Secretary of State...