Lynch is a family name; William Lynch is apparently the point of its association with hangings. As the linked site indicates "William Lynch (1742-1820) of Pittsylvania, Virginia, who c.1780 led a vigilance committee to keep order there during the Revolution."

Communities bearing the name "Lynch" are likely named after someone with the last name or after a tradition of lynching. I notice that the origin of the Tennessee community is unclear, having attracted, apparently, several folk etymologies.

"Lynch" does not mean the racial killing of a black person; it generically refers to the extra-judicial or non-judicial killing of a person usually by a mob. It just happens that in American history a lot of lynchings are racial. But the term is not exclusive to racially-based incidents.