The band The Rolling Stones did not derive its name from the song Like A Rolling Stone by Bob Dylan.
According to Keith Richards, it was the former guitarist Brian Jones who came up with the name in a rather accidental way. Brian glanced down at a record while being asked for a name, which happened to be Muddy Waters’ Rollin’ Stones. As a result, he opted for this name, and it stuck incredibly well.
The song Like A Rolling Stone by Bob Dylan was released in 1965, three years after the band The Rolling Stones was formed in 1962. The title of the song is not a reference to The Rolling Stones, but to the proverb a rolling stone gathers no moss which means a person who does not settle in one place will not accumulate wealth or responsibilities.
Dylan got the idea from the 1949 Hank Williams song Lost Highway, which contains the line, I’m a rolling stone, all alone and lost.