Levi A. Weaver was born on April 22, 1833, in Philadelphia, Pennsylvania, to Levi and Elizabeth Weaver. His father was a teamster who owned $5,500 of real estate and $1,200 of personal property by 1860. The family moved to Harrisburg, Pennsylvania, around 1838, and he grew up and attended school there. At the age of 13, he began a three-year apprenticeship as a store clerk. After completing his apprenticeship, he helped his father operate a boat on the Pennsylvania Canal. He moved to Texas around 1858, but he returned to Pennsylvania by 1860.
In August 1861, he received a commission as a 2nd lieutenant in Company G of the 55th Pennsylvania Infantry. The regiment took part in the Battle of Cold Harbor and the siege of Petersburg. He was promoted to 1st lieutenant in May 1863, and to captain in July 1864. He was wounded in May 1864, and he mustered out on November 23, 1864. In April 1865, he returned to the Union army, mustering into the 7th United States Veteran Volunteer Infantry. They performed guard duty in Washington, D.C., and Philadelphia, Pennsylvania, and he mustered out in April 1866.
He returned to Harrisburg after the war, and he worked in the railroad industry. He married Mary J. Lukens, and they had at least two children. In February 1892, he developed rheumatism, which forced him to retire. He died in Harrisburg on April 23, 1899.