I'd prefer "Joe's plan". After that, "a plan of Joe's" works, although it's a little colloquial and possibly regional. It works best when "plans that Joe has" are a reasonably usual occurrence. You can think of it as short for "a plan of Joe's devising."
"A plan of Joe" is technically correct but it unusual, jarring, and (to some) unpleasant to read.
Btw, "a cup of Joe" means "a cup of coffee" so that might be why I find that third option so odd. :)