Laporta, who held the top job between 2003 and 2010 and is one of the frontrunners along with incumbent Josep Maria Bartomeu, presented Abidal at a news conference in the Catalan capital on Monday.
Abidal, 35, played for Barca during a glittering run for the club between 2007 and 2013 and became a fan hero when he made a successful return after undergoing a liver transplant in 2012.
He joined Ligue 1 side Monaco in July 2013 and announced his retirement as a player in December after a brief stint with Greek club Olympiakos Piraeus.
"Barca is more than a club and I wanted to carry on working and maintain a link to that," Abidal, who would help oversee Barca's transfer market dealings, told reporters.
Another of his tasks would be to ensure Barca's La Masia academy, which produced the likes of Lionel Messi, Andres Iniesta and Xavi in recent times, remained the centre of excellence, he added.
A lawyer and pro-Catalan independence politician, Laporta has tried to distinguish his candidacy from Bartomeu's by criticising the sponsorship deal Barca signed with Qatar under former president Sandro Rosell.
Rosell took over from Laporta in 2010 but was forced to resign in January 2014 after he and the club were accused of tax fraud in the signing of Brazil forward Neymar.
Bartomeu stepped up to the top job and has decided to run for another term even though he has also been named as a defendant in the case.
As part of the deal with Qatar, Barca scrapped their policy of not displaying the name of a shirt sponsor and relegated the logo of long-term collaborator Unicef to the back of their jerseys, angering many of the club's members.
Laporta suggested the Unicef logo would be restored to the front of the shirt in place of current sponsor Qatar Airways but did not say how he might make up for the lost revenue of some 30 million euros (21.49 million pounds) a season.
"We are for Unicef and they are for Qatar," Laporta, who is close to former Barca player and coach Johan Cruyff, said.
"Having money is important but it is not everything in the world. Principles come before money."
(Reporting by Iain Rogers, editing by Sudipto Ganguly)
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