No, one was not named after the other, both were in fact named after Muddy Waters' song, Rollin' Stone. Recorded in 1950, Rollin' Stone is Muddy Waters version of a traditional blues song, Catfish Blues. By the way, Muddy Waters is a person and not a group. While the song didn't chart, it was commercially successful and Muddy was able to quit his full time job.
The Rolling Stones on the other hand, were formed in 1962 in London - four years after Muddy had toured the country. They released their first album in 1964, composed mainly of covers, and toured the States that same year. Since then the band has released 29 albums and over 100 singles.
Rolling Stone was founded in 1967. Not just about music, the magazine is also "about the things and attitudes that music embraces." Rolling Stone was noted for its liberal politics in the 1970s and featured Hunter S. Thompson's gonzo writings including Fear and Loathing in Las Vegas in 1971.
Anyway, without further ado, lets travel back in time to 1960 and visit the traditional home of the America's Cup, Newport. On today is the Newport Jazz Festival and here is Muddy Waters playing Rollin' Stone.