The 'R' stands for nothing. His birth name was JR Cash not Johnny Cash.
Explanation:
When Johnny Cash was born his parents couldn't decide on a name for him so they just went with JR. His real name is JR.
JR are technically initials and not an actual name which caused a problem for him when he wanted to enlist in the airforce, initials were not allowed. Therefore he enlisted himself as John R Cash. He later changed it to Johnny Cash as he gained popularity in the music world.