I'm surprised that so many people are questioning Weber's induction. It shows the flaw in counting trophies as a binary (yes or no) outcome. Weber was consistently one of the best defensemen in the league for close to a decade.
There's a big difference between someone who repeatedly comes close to winning a trophy, and someone who was never close at all (ie Kevin Lowe, Leo Boivin). There are plenty of great defensemen who never won a Norris but at least came close on multiple occasions (Brad Park, Borje Salming, Scott Stevens, Tim Horton, etc). There's plenty of precedent for Weber being inducted.
The simplest argument? Weber placed in the top five for the Norris trophy 5 times in his career. 24 other defensemen in NHL history have done that. Every single one that's eligible is in the Hall of Fame. (The only "exceptions" are Erik Karlsson and Victor Hedman, who are still active, and who will obviously be inducted in the future).
From 2009 to 2017 (a span of nine seasons), Weber finished 3rd in the NHL in votes for the Norris trophy (normalized so each year is worth the same number of votes) - behind Karlsson and Chara, but ahead of some big names like Doughty and Keith. During his peak (2011 to 2015 - five season), even though he didn't win the Norris, he was easily the leader in votes during that period (ahead of Chara, Karlsson, Subban and Keith).