As the granddaughter, daughter, sister and mother of five Washington Post publishers, Ms. Weymouth moved in social circles that helped shape American media and government, from the Camelot age of John F. Kennedy to the dawn of the Trump era more than 50 years later. As a young girl, she attempted to teach her mother, Katharine Graham, the twist in preparation for dance parties at the White House. She later swapped stories about her father, Philip L. Graham, with former president Lyndon B. Johnson, a family friend. To celebrate her mother’s 70th birthday, she organized a glittering 600-person party that...