I would prefer the Hall of Fame over the Ring.
But only over the Ring itself. Even though I think the Hall is the greater honour compared to being one part of a winning team - because this always also is a bit of luck -, I would prefer to win the finals, the Super Bowl, or for me the Stanley Cup, - nothing in sports is better than the moment you win the last game and take the championship home. Even as a fan it's so fucking awesome.
Next choice, and please think of the trophy itself, not of the sport or event: Super Bowl Vince Lombardi Trophy or Stanley Cup?