I’ve read that the magazine took its name from the Bob Dylan song, Like a Rolling Stone.
However the site where I got the information from also repeated the old “Frank Zappa is the son of Mr. Greenjeans on Captain Kangaroo” story so…
What I hear is that both are named after an old blues song. No direct correlation between them.
Here’s what I believe both magazine and band named themselves after. It’s a Bo Diddley/Muddy Waters song.