Lawrance was born on September 30, 1882 in Lenox, Massachusetts, the son of Francis Cooper Lawrance Jr. (1858–1904) and his first wife, Sarah Eggleston Lanier (1862–1893). After his mother's death in 1893, his father remarried to Susan Ridgeway Willing, a sister of Ava Lowle Willing (who married John Jacob Astor IV). They had a daughter, a half-sister to Lawrance, Frances Alice Willing Lawrance, who married Prince Andrzej Poniatowski of the House of Poniatowski in 1919. From his parents marriage, Lawrance had a younger sister, Kitty Lanier Lawrance, who was raised by their paternal grandfather, as their parents died when she was still young. In 1915, Kitty married W. Averell Harriman, the Governor of New York (they divorced in 1928).