Ovid, New York. Ovid is a village in and one of the two county seats of Seneca County, New York, United States. The population was 602 at the 2010 census. The town was named by a clerk interested in the classics (see Ovid). The Village of Ovid is within the Town of Ovid, but a small portion is in the Town of Romulus, and is southeast of Geneva, New York.