Chris Skura followed up and told us that the band came first (1962), five years before the magazine (1967). But that both were named after the song “Rollin’ Stone,” written and recorded by Muddy Waters in 1950.
So where does that leave Bob Dylan’s song “Like a Rolling Stone” that came out in 1965, three years after the band and two years before the magazine? Well, Jann Wenner – founder of Rolling Stone – has reportedly said that he named the magazine after the propular expression “a rolling stone gathers no moss,” and perhaps Dylan had the same expression in his head when writing his breakout hit. Interestingly, the magazine at one point named Bob Dylan’s “Like a Rolling Stone” and ”Satisfaction” by the Rolling Stones as the two top rock songs of all time.
And, finally, a note from Wikipedia: The television show MythBusters, over the course of six months, confirmed that a rolling stone does not grow moss.