Victorian-born Adcock was co-captain alongside Brown in 2013, but the 28-year-old will become standalone skipper in 2014.
The new hierarchy was settled earlier this week during a senior player leadership conference in Noosa.
Midfielder Tom Rockliff was named the club’s vice-captain, while Daniel Rich, Joel Patfull and Daniel Merrett will take up roles as deputy vice-captains.
Brown, 32, strongly considered retiring at the end of the last season.
"Browny's role will be to mentor our leaders this year. It was a joint decision," Leppitsch said.
"It will help (Brown) personally but I think it will help our club develop new leaders.
"If Brownys in those meetings, he's always heard and always dominant.
"It's time the next generation started to come through."
Brown supports the changes and also wants fresh faces to lead the Lions, who have not made the finals since 2009.
"Obviously I have relished my time as captain, but I think it's the right time for a changing of the guard," Brown told the Lions website.
"I'm coming towards the end of my career and I think with Leppa on board it's the perfect time ... I'll still be there to support the boys when they need, but I have every confidence in the ability of Jed, Tom and our next generation of leaders to take the club forward."