What is the meaning of the expression "A one paper kid" or "One paper kid"

Given that the song lyrics contain references like light it again and rollin' again, the paper in question is likely cigarette paper.

As for the meaning of the expression, another phrase in the lyrics says "times were so lean," so it's likely a metaphor for that -- the "kid" (some Western connotations there as well) is down to his last cigarette paper, or economizes by rolling cigarettes with only one paper.

In the 70s, many people stuck two regular-size rolling papers together to make a larger one, to make rolling a joint easier. Using one paper was a mark of skill and experience.

Rolling a joint with one cigarette paper was a sign of virtuosity even before the 70s. Anyone who spent much time rolling joints would recognize this allusion right away.