It is the second time the hooker has skippered the Lions during the tour, having led them to a win over the Chiefs.
Coach Warren Gatland made the decision after the Lions lost the opening Test against New Zealand 30-15 on Saturday.
The second Test against the All Blacks is on Saturday, 1 July.
"We are obviously hugely disappointed with the result but it is important to regroup," said Gatland.
"We know there are a lot of players motivated to go out and get a result for the whole group.
"A few players put their hands up for selection from the Chiefs game and played their way into the Test team so they know that there is another opportunity for them on Tuesday against the Super Rugby champions."
Wing George North and centre Jonathan Joseph will both start against the Hurricanes, while Jack Nowell is at full-back.
Lock George Kruis is on the bench having played the whole 80 minutes of the first Test against New Zealand.
The All Blacks have released Julian Savea, Jordie Barrett and Ngani Laumape back to the Hurricanes for the game, while Nehe Milner-Skudder also starts.
British and Irish Lions: Jack Nowell (England), Tommy Seymour (Scotland), Jonathan Joseph (England), Robbie Henshaw (Ireland), George North (Wales), Dan Biggar (Wales), Greig Laidlaw (Scotland), Joe Marler (England), Rory Best (Ireland, captain), Dan Cole (England), Iain Henderson (Ireland), Courtney Lawes (England), James Haskell (England), Justin Tipuric (Wales), CJ Stander (Ireland).
Replacements: Kristian Dacey (Wales), Alan Dell (Scotland), Tomas Francis (Wales), Cory Hill (Wales), George Kruis (England), Gareth Davies (Wales), Finn Russell (Scotland), Jared Payne (Ireland).
Hurricanes: Jordie Barrett, Nehe Milner-Skudder, Vince Aso, Ngani Laumape, Julian Savea, Otere Black, Te Toiroa-Tahuriorangi, Ben May, Ricky Ricitelli, Jeffrey To'omaga-Allen, Mark Abbott, Sam Lousi, Vaea Fifita, Callum Gibbins, Brad Shields (captain).
Replacements: Leni Apisai, Chris Eves, Mike Kainga, James Blackwell, Reed Prinsep, Kemara Hauiti-Parapara, Wes Goosen, Cory Jane.
Rory Best will captain the British and Irish Lions for his side's next game against the Hurricanes in Wellington on Tuesday.