Monroe is a town in Turner County, South Dakota, United States. The population was 159 at the 2020 census. Monroe was originally called Warrington, and under the latter name was platted in 1887. Some say the present name is for James Monroe, fifth President of the United States, while others believe the name is a transfer from Monroe, Wisconsin, t…Monroe is a town in Turner County, South Dakota, United States. The population was 159 at the 2020 census. Monroe was originally called Warrington, and under the latter name was platted in 1887. Some say the present name is for James Monroe, fifth President of the United States, while others believe the name is a transfer from Monroe, Wisconsin, the native home of an early settler. A post office called Monroe was established in 1890, and remained in operation until it was discontinued in 1965.