Alexander Hamilton Davidson was born in Anson, North Carolina, in 1817. One of six children, he was the son of Sidney Green Davidson (1783–1859), born in Montgomery County, North Carolina, and Delphia Atkins (1783–1878) of Anson, North Carolina. His parents married in 1799 and died in Dyer County, Tennessee. Davidson's first wife, Eliza Jane Lott (1818–1848), was the mother of his eldest son William Lott Davidson, who was born in Mississippi in 1836. After Eliza's death in 1848, Davidson married Ruth Tennessee Ragsdale (1831–186?) in Monroe County, Mississippi, in 1852. They had four children. Their last three children were all born in Texas. The Davidsons moved to Texas in the mid-1850s and settled in Colorado County. Alexander Hamilton Davidson was a wealthy and important member of Colorado County society. He was an attorney and landowner, and he owned nineteen slaves in 1860. Before moving to Colorado County, Davidson owned an 867-acre plantation on the Medina River in Bexar County, which he sold on April 23, 1860. He then set up a plantation south of Columbus, Texas, on Skull Creek.
Davidson was considered a Cooperationist in 1860, not being in favor of secession. He was part of a committee that drafted a resolution for approval by the assembly of Colorado County on the issue of whether Texas should secede from the Union. Davidson supported a document that called for Sam Houston to run for president, denied that any state could secede from the Union except by revolution, and declared that there was not sufficient cause for a revolution. The resolution against secession failed, however, and Davidson soon became a supporter of disunion.
Davidson was elected a delegate from Colorado County to the Texas Secession Convention and attended its meeting in Austin in January-February 1861. Davidson, along with Thomas Scott Anderson of Colorado County, voted in favor of the ordinance of secession.
After the war broke out, Davidson secured a commission in the Confederate Army as a lieutenant colonel of the First Texas Cavalry Battalion and set out to raise a battalion to serve in the New Mexico campaign. The troops he initially raised were re-assigned to another commander though, and he was left in Texas. In October 1862, with the help of his son William Lott Davidson and Samuel G. Ragsdale, he raised a battalion.
Davidson's battalion was originally known as Texas First Cavalry Battalion, Arizona Brigade, and he was the first commander. In September of 1863, the battalion was ordered to Galveston to serve as dismounted infantry, and a month later in October, Davidson was killed in action in Louisiana. Although there is no definitive evidence, it is highly probable that he died at Bayou Bourbeau on November 3, 1863. The battalion became known as Ragsdale's Battalion, Texas Cavalry, after Samuel G. Ragsdale took over the command following Davidson's death.