The band got their name from an old saying, "A rolling stone gathers no moss." (It was also the title of a 1950s Muddy Waters song.) This saying has a literal truth in that moss needs an unmoving environment in order to form. One of the easiest ways to get an idea of how long stones or trees have been in a place is to look at the amount of moss that has formed on them. As a proverb, it can be taken in two ways.
The basic idea, that if you keep moving (never settle down) you won't accumulate anything, is usually taken to mean that you will never amass any wealth or have a settled career. The other possible meaning is that you will never have any responsibilities such as home or family, but will remain "clear" and unencumbered. Either way, the saying associates the idea of a roving life with both freedom and a certain lack, be it of possessions or relationships.
The basic idea, that if you keep moving (never settle down) you won't accumulate anything, is usually taken to mean that you will never amass any wealth or have a settled career. The other possible meaning is that you will never have any responsibilities such as home or family, but will remain "clear" and unencumbered. Either way, the saying associates the idea of a roving life with both freedom and a certain lack, be it of possessions or relationships.