If you found a magical stone that could keep one person young forever, would you keep it or give it to someone else? If the latter, to whom would you give it?
ANSWER #1 (real answer):
I'd drop it in the ocean, or a volcano, or launch it into space, or something like that. Can you imagine the conflict a stone like that would cause if anyone found out about it? Better that everyone gets to live their full life (even if it seems short), than a bunch of people get hurt/killed fighting over who gets to live young forever.
ANSWER #2 (answer that would make a good story)
I'd use the stone to live as long as possible. If I ever died of something other than old age, I'd pass the stone on to some chosen "heir". With virtually unlimited time on my hands, I'd be able to work to become the most intelligent and skilled person in the world. Eventually (since I have practically forever to figure out how to make this happen), I would find a way to take over the entire world, and rule it as a wise and benevolent monarch.
(Okay, I wouldn't really do that -- it's just what'd make the best story. And if anyone wants to write a story with that premise, feel free.)