Perhaps Finny and the other boys call their club "The Super Suicide Society of the Summer Session" in order to impress others with their daredevil antics, but Gene originally names it "The Suicide Society of the Summer Session" because of his fear.
After Phineas and the other boys have attended the "traditional term tea for the Upper Middle class held in the "deserted Headmaster's house where Mr. Patch-Withers attends the boys," Finny decides that he and Gene should "go jump in the river" in order to clear their heads after the party that was "all talk." As they swim in the river, Finny asks Gene, "Are you still afraid to jump out of the tree?" and Gene notes something "unpleasant" about the question, but Finny replies that it depends upon how he answers the question. So Gene feigns confidence. Then, when Finny asks him to do him "the pleasure of jumping out of the tree first," Gene tersely replies, "My pleasure." But, he is "rigid" as he climbs the rungs of the ladder. His rigidity, his fear, is what drives Gene--after hearing Finny declare,
"We'll jump together to cement our partnership...We'll form a suicide society, and the membership requirement is one jump out of this tree"
--to say stiffly, "A suicide society....The Suicide Society of the Summer Session." Finny loves the name. He alters it to "The Super Suicide Society of the Summer Session."
In fact, the "jump" that Gene was to make could have been suicide because he slipped on the branch, and if Finny had not caught him, he could have broken his back, or even been died.