Senator Clinton’s closest aide, Ms. Abedin never sweats; Oscar de la Renta wants to dress her Last June, under an oppressive sun, at a rally to save the Niagara military base at the University of Buffalo, all of New York’s top politicians—George Pataki, Chuck Schumer, Hillary Clinton—poured sweat. Yet there was exactly one member of the wilting delegation who managed, somehow, to stay cool: Hillary Clinton’s mysterious, glamorous and eerily unflappable aide de camp, Huma Abedin. “It was like 110 degrees outside,” recalled the source, a political aide who asked to remain anonymous. “We were all just pouring down with...