The Super Suicide Society of the Summer Session is the name that Gene and Finny create together in the course of jumping from the tall tree by the river, an activity that Gene deeply fears and detests:
Rigid, I began climbing the rungs, slightly reassured by having Finny right behind me. "We'll jump together to cement our partnership," [Finny] said. "We'll form a suicide society, and the membership requirement is one jump out of this tree."
"A suicide society," I said stiffly. "The Suicide Society of the Summer Session."
"Good! The Super Suicide Society of the Summer Session! How's that?"
"That's fine, that's okay."
This exchange reveals a lot about each boy. Finny uses the word "suicide" lightly; he fears nothing about what they are doing, welcoming the challenge. Gene, however, is petrified with fear. His body is rigid and he takes no joy in the cleverness of the title they have come up with. For Gene, climbing the tree and jumping from it does seem like an act of suicide. The first time he had jumped from the tree, Gene had done so "[w]ith the sensation that [he] was throwing [his] life away."
One way to examine this question is to consider how jumping from the tree ultimately leads to Finny's death. While Finny survives the fall from the tree when Gene knocks him out, the broken leg Finny suffers almost directly leads to his death later when he falls again and complications from this break ensue.
The brazen idea of forming a Super Suicide Society of the Summer Session thus proves to be Finny's undoing. Before he dies, however, Finny's whole vision of his future is also destroyed.
Where he had wanted to try out for the Olympics (and realistically probably could have done so - and made it), his broken leg erases any chance at making an Olympic team. Where Finny's identity had been largely located in his physical prowess, crutches and the sullen, evasive attitude that Finny is saddled with during his recovery shifts his identity into an entirely new place. He is no longer the charming and winning personality he once was.
These results of Finny's fall from the tree are made somewhat ironic when compared to the first jump of the Suicide Society. At that jump, Gene nearly falls out of the tree but Finny saves him.
"If Finny hadn't come up right behind me...if he hadn't been there...I could have fallen on the bank and broken my back! If I had fallen awkwardly enough I could have been killed. Finny had practically saved my life."
This idea recurs toward the novel's end, when Gene attends Finny's funeral. Blaming himself as the cause of Finny's death and identifying so fully with Finny at this point (after acting as Finny's stand-in for Olympic training, etc.), Gene reflects on Finny's death in a way that brings his story back to his own death.
"I could not escape a feeling that this was my own funeral, and you do not cry in that case."
The "true Finny" dies when Gene knocks him from the tree, but Gene's life is already somehow indebted to Finny and identified with Finny's from that first jump when Finny saves him. Thus, when Finny dies and is buried and Gene feels that it is his funeral too, the "suicide" is double.
Although there was no intention of actually killing themselves, the recklessness of the central activity for the Super Suicide Society proves to be the mechanism for this twin death of Gene and Finny, ending a time of innocence for each well before their lives end, actually and metaphorically.