Thayer is a city in Oregon County, Missouri, United States. The population was 1,883 at the 2020 census. Its original name was Augusta. A pos…Thayer is a city in Oregon County, Missouri, United States. The population was 1,883 at the 2020 census. Its original name was Augusta. A post office called Thayer has been in operation since 1884. The community has the name of Nathaniel Thayer, a railroad promoter.