When Chelsea Clinton wanted to make a low-key visit to the hurricane-stricken Rockaways last fall, she arranged to take a trip with her close friend Huma Abedin, a longtime aide to Hillary Rodham Clinton. -snip- But the reality, it turns out, is just the opposite: Those close to the couple say that Ms. Abedin, a seasoned operative well versed in the politics of redemption, has been a main architect of her husband’s rehabilitative journey, shaping his calculated comeback and drawing on her close ties with one of the country’s most powerful families to lay the groundwork for his return. A...