Conlon and the rest of the Guildford Four served 14 years of a life sentence for the attack, in which five people died and 65 were injured, before their convictions were overturned in 1989.
The case was highlighted in the film In The Name Of The Father, in which Conlon was played by Daniel Day-Lewis.
His family issued a statement through his lawyer, Gareth Peirce.
It said: "This morning we lost our Gerry. He brought life, love, intelligence, wit and strength to our family through its darkest hours.
"He helped us to survive what we were not meant to survive. We recognise that what he achieved by fighting for justice for us had a far, far greater importance - it forced the world's closed eyes to be opened to injustice; it forced unimaginable wickedness to be acknowledged; we believe it changed the course of history. We thank him for his life and we thank all his many friends for their love."