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).
Multiple things, but I think thosep lay a big role in the perception of Weber's career today:
1. Recensy bias: Offense is high, more than 1 defensemen breaking pts per game this year, something that was considering almost a HHOF guarantee if done during Weber's peak years. His pts totals don't look that impressive in the context of today's offense, but he was consistently top 10-20 in D scoring.
2. His biggest asset was defense. Nothing else to day, defense is difficult to judge retroactively.
3. He played for Nashville during his prime/peak, who were a heavy defense-first team during his tenure there. Little flash, little media spotlight.
4. He then joined Montreal... who were not super competitive either. I think, if by miracle, they got that Cup, the narrative changes drastically. But they didn't, and not being a lifer on Nashville might actually have hurt his legacy amongst fans? As odd as it sounds
I know it's just math and not perfect metrics, but if you look at Hockey-reference's career similarity scores, his closest comparables include guys like Chara, Doughty, Niedermayer, Chelios, Pronger, all HHOF or to-be HHOF members