Lea was the son of Overton Lea and Ella (Cocke) Lea. He was born into a political family after Reconstruction and named for a paternal great-grandfather, Luke Lea, who was a two-term Congressman from Tennessee in the 1830s. Initially an ardent supporter of Democrat Andrew Jackson, the elder Lea later became a member of the Whig Party. One of Lea's maternal great-grandfathers was William Cocke, who served in the U.S. Senate from Tennessee from 1796 to 1797, and again from 1799 to 1805.