The term that you are looking for here is “grandfather clause.” It is a colloquial term that is used because of the fact that, as you mention, people were allowed to vote if their grandfathers had been eligible to vote at the time of the Civil War. The point of these clauses was to allow white people to vote while preventing African Americans from doing so.
After the Civil War, the 15th Amendment to the Constitution was ratified. This said that the right to vote could not be abridged on account of race. This meant that Southern governments had to find other ways to exclude blacks from voting. Therefore, they instituted things like literacy tests and poll taxes to prevent uneducated and/or poor blacks from voting. The problem was that these things also prevented many white people from voting. The legislatures therefore needed to find a way to make sure that those white people would be allowed to vote. The grandfather clause was one of these. It stipulated that people who failed the literacy tests, for example, could still vote if their grandfather had been able to vote. This allowed whites to vote but excluded blacks without actually officially discriminating against them on the basis of race.