I know that he is a former coach and executive. Keeping it under his name and having someone look it up helps teach the history of hockey to those who would be interested.
Under the same argument, if I would look up Bergeron in 30 years, I would see he is a player that won the award a lot. He didn't change the way the game was played in anyway. I would not learn any history of the sport other than he won one trophy a lot, which I would have actually been able to see if I looked up the winners of the Selke at some point.
So lets say we name it after Patrice, when do we change it? Do we go by who wins it the most? What about when their is a tie for who has won it the most? Do we need to name it with both? For the yearly Messier award, that means it is named after 16 people now. Calder and Masteron are an even bigger problem. Are we in a better spot with the Lady Bing being named after Frank Boucher? Is the Vezina better as the Plante Award? The Jack Adams would be named after Pat Burns, who from all accounts was a lovely person and a great coach, he also only won one Stanley Cup. I think there are at least 5 coaches that you would put on there before him.
If we go because Bergeron is the most current really good defensive forward, when do we change it? Why are we going off of current as oppose to players such as Fedorov or Datsyuk who are fairly recent. Heck, Fedorov was good enough that his coach played him at defense.
It is creating a solution to something that isn't a problem to try and give unneeded extra recognition to a player for no real reason. Simply saying, "Bergeron has won the Selke for the best defensive forward more than anyone" is enough to say that "Hey, this guy was a really good defensive forward".