Lamesa, Texas. Lamesa (/ləˈmiːsə/ lə-MEE-sə) is a city in and the county seat of Dawson County, Texas, United States. The population was 9,422 at the 2010 census, down from 9,952 at the 2000 census. Located south of Lubbock on the Llano Estacado, Lamesa was founded in 1903.