As others have pointed out, "famous" is subjective and a bit ephemeral. I would like to adjust the question to ask for mathematicians who have actually contributed more to mathematics by their conjectures than via their theorems. This is still subjective, but distinctly less so.
Anyway, when I try to think of a mathematician whose greatest contributions lie with his conjectures and for which those contributions are enormous, one name springs to mind: Robert Langlands. The Langlands program is one of the most important and influential pieces of 20th and 21st century mathematics. I would imagine that Langlands himself would agree that his program is (even) more important than the results he has proved: in fact, most or all of his important results feed into and elevate his program.
The OP says
I am interested to explore whether some mathematicians have specific conjecture-talent that is not evidently reflected in theorem-proving talent.
I think this is a good example of this, of a certain particular kind: Langlands clearly has remarkable theorem-proving talent. However his conjecture-talent is beyond remarkable...it is Langlandsesque.
I think there is a further small perturbation of the question which makes the canonical answer Paul Erdős. If we rank mathematicians by number of theorems proved then Erdős surely comes near the top of the list. However, the influence that he has had on contemporary mathematics goes beyond any one result of his. Erdős died (almost exactly) 20 years ago. That was right about the point where I started paying attention to the mathematical landscape, in particular number theory. The last 20 years have been a stampede towards the kind of problems that Erdős proposed. In particular, the amount of leading mathematical work done in that time towards the Erdős–Turán conjecture alone is enormous.