The quest for a first championship for Weber and goalie Carey Price is being used as a rallying point by Canadiens players. Weber has been in the NHL since 2005, and Price entered the League in 2007.
"This is for them at this point," rookie forward Cole Caufield said.
For Weber, it has been a bumpy trip to get here. He spent his first 11 seasons with the Nashville Predators before he was traded to the Canadiens on July 29, 2016, for P.K. Subban, the winner of the 2013 Norris Trophy voted as the best defenseman in the NHL and one of the more popular players in the community.
General manager Marc Bergevin said he always had confidence in Weber's influence on the team, even though many Canadiens fans were not on his side at the time.
"He came here as part of a trade where P.K. was traded for 'Webby,'" Bergevin said Sunday. "And I know there's a lot of people that were guessing or second-guessing this transaction. And again, to get something that special you have to give up a good player. So I'm not taking anything away from P.K. But to get Shea Weber here in Montreal, and what he brings on and off the ice for me, it's special. And that we're four wins away from winning a championship will be ultra-special."
Bergevin has had fun with the decision. In April, the GM was featured in a commercial for Weber barbecues that ends with him saying in French: "I've always known Weber was the right choice." The spot went viral in Montreal.
Injuries have been Weber's biggest issue with the Canadiens, causing him to miss 84 games in his first three seasons in Montreal. The most serious was a tendon tear in his left foot that limited him to 26 games in the 2017-18 season.
Weber missed the final eight games this season with an upper-body injury but has played in all 17 playoff games. The 35-year-old has four points (one goal, three assists) and leads all players participating in the Cup Final in average ice time per game at 25:38.
The hardships of the past have caused Weber to embrace what lies ahead.
"I've been really enjoying my time here," he said. "Been fortunate to be traded to a great city and organization. And yeah, there's been some hard times, there's been some good times, but it's a part of the journey and a part of the story.
"It's been a good ride so far and hope to keep it going here."
Weber's physicality has been a key for the Canadiens in the first three rounds, especially when it comes to limiting second-chance opportunities on Price. He's used his 6-foot-4, 229-pound frame to box out opposing forwards and allowed the Canadiens goalie to more often than not get a clear view of shots coming in.
It's a recipe he'll have to maintain against a Lightning team that sports elite offensive firepower from talented forwards like Nikita Kucherov, Steven Stamkos and Brayden Point.
Canadiens defenseman Ben Chiarot said Weber is up to the task and is the perfect example of a player who leads by example.
"I think the thing with Webby is once we hit the ice it's all business," Chiarot said. "There's no messing around. He wants everything done exactly the way the coaches said it's done. And I think that's a big reason why he's been so successful as a player, his attention to details, his seriousness once we hit the ice.
"And off the ice, he's got a relationship with everybody in the room. The young guys, he's dad to the young guys, and he's buddies with all the older guys. He connects with everybody. And I think that's why he's considered one of the best captains in the League."