Some years after the death of Dr Joseph Cleburne, the Cleburne children came to America. Patrick arrived in New Orleans Dec 25, 1849 with sister Anne (age 23) and brothers William (age 25) and Joseph (age 19). All went upriver to Cincinnati OH. Patrick soon found an opportunity in Helena AR, leaving his brothers and sisters in Ohio. In 1850 William and Joseph are listed on the census as civil engineers living in a boarding house in Shelby County OH. In the 1880s Anne and William Cleburne donated funds to the memorial planned in Helena AR. I don't know where Anne or Annie Cleburne lived or whether she had married at that time. Evidently in 1860 she was living in Cincinnati and had a family.
Joseph Cleburne had long since disappeared by that time.
The Arkansas Historical Association makes reference to a sister who lived in Omaha NE. This is a mistake. Patrick R Cleburne had a older brother named William who lived in Omaha. After the war William came from Indiana to work as an engineer for the Union Pacific Railroad. In 1870 William had two daughters named Minnie and Laura who had been born in Indiana. He appears on the 1880 census as Joseph Cleburne, which may have been his middle name. (Dr Joseph Cleburne was his father).
William Cleburne appears on the Omaha city directory until 1904, at which time he would have been about eighty (born Ireland in 1824). Later in life he became a noted Nebraksa naturalist.
In 1856 Patrick R Cleburne helped his stepmother Isabella and her children come to America. They joined Anne Cleburne and her family in Cincinnati. Cleburne's younger half-brothers and half-sisters were Robert, Christopher and Isabella. Half-brother Edward died at sea in 1853; Christopher died in Confederate service at the Battle of Cloyd's Mountain VA, May 10, 1864. Robert and Isabella moved to Maysville KY before 1860 and continued to live there. All three appear on the 1880 census of Kentucky. Robert Cleburne died on Apr 26, 1905, and is buried at Newport KY.
A town in Texas was named Cleburne, and many Confederate veterans who served under General Cleburne lived there in the post-war years. Other than that I can find no connection of Texas and Patrick Cleburne's family.
Resources -- Mauriel Phillips Joslyn, A Meteor Burning Brightly: Essys on Maj Gen Patrick Cleburne. For relevant family data Ms Joselyn cites Howell and Elizabeth Purdue, Pat Cleburne, Confederate General: A Definitive Biography.