Elgin is a census-designated place in Santa Cruz County, Arizona, United States. The population was 161 at the 2010 census. Elgin is located along the Babocomari River at 31°37′45″N 110°34′19″W / 31.62917°N 110.57194°W. The Babocomari is known to harbor a unique species of fish, belonging to the family Characidae. They are incredibly rare an…Elgin is a census-designated place in Santa Cruz County, Arizona, United States. The population was 161 at the 2010 census. Elgin is located along the Babocomari River at 31°37′45″N 110°34′19″W / 31.62917°N 110.57194°W. The Babocomari is known to harbor a unique species of fish, belonging to the family Characidae. They are incredibly rare and little is known about them. According to the United States Census Bureau, the CDP has a total area of 5.9 square miles, all land.