In July 2000, Hillary Clinton stood on a stage at the University of Arkansas and struggled to keep her composure. Her voice was unusually soft that day, her words seemingly unfiltered. “Diane, you were the awesomest,” Mrs. Clinton said, referring to Diane Blair, a political science professor whose eulogy she was giving. “You were the best person that one could have as a friend.” Mrs. Blair, who died at 61, was described as the sister Mrs. Clinton had always wanted. She practically moved into the White House in 1993 to ease Mrs. Clinton’s transition to Washington, and returned at the...