Asa Miner Weston was born on September 24, 1836, in Ohio to Asa and Thankful Weston. He was baptized in the First Presbyterian Church in Cleveland, Ohio, on November 6, 1836. He probably grew up and attended school in Cleveland before enrolling at Oberlin College.
He enlisted in the Union army on August 11, 1862, and he mustered in as a sergeant in Company K of the 50th Ohio Infantry later that day. The regiment took part in the Battle of Perryville, the Atlanta campaign, and the Carolinas campaign. He supported President Abraham Lincoln in the election of 1864. He was promoted to sergeant major on March 4, 1865, then to 2nd lieutenant on April 22, 1865. He mustered out on June 26, 1865.
Weston returned to Ohio after the war, and he married Julia Pardee there on July 7, 1868. Their daughter Nellie was born around 1869. By the late 1860s, he was working as a professor of mathematics and modern languages at Hiram College. The family moved to Jefferson, Indiana, in the 1870s, and he worked as a farmer there. By 1900, they were living in Henry, Indiana. He died there on November 11, 1902.