In Deborah Brown’s family lore, the American South was a place of whites-only water fountains and lynchings under cover of darkness. It was a place black people like her mother had fled. Multimedia Deborah Brown, 59, and her mother, Louise Brown, 82, plan to move from Queens to Atlanta. But for Ms. Brown, 59, a retired civil servant from Queens, the South now promises salvation. Three generations of her family — 10 people in all — are moving to Atlanta from New York, seeking to start fresh economically and, in some sense, to reconnect with a bittersweet past. They include...