(from Cambridge Dictionary of American English)

stiff (PERSON)
noun [C]
SLANG
a person of the type described
I'm just a working stiff.
You lucky stiff!
(SLANG) A stiff is also a dead person's body.

Looking at various dictionaries it seems that a 'stiff' is slang for an ordinary person, usually used in conjunction with 'working' or 'lucky'. No indication of the origin is given anywhere that I can find.