What is the significance of "the super suicide club" in A Separate Peace?

The "Super Suicide Society of the Summer Session" has a light-hearted name and is a "success" from the start, but from its inception Gene knows the seriousness of what it asks of its members. Jumping out of the tree, even once, into the river, which Finny decides is the minimum requirement for membership, is a risky undertaking. As Gene understands, only Finny grabbing his arm saved him from a fall that might have caused a serious injury the first time he jumped:

It was only after dinner, when I was on my way alone to the library, that the full danger I had brushed on the limb shook me again. 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!

Of course, knowing this, Gene will nevertheless bounce the tree later, causing Finny's terrible fall.

The name is also significant because other students assume it is one of the old, fabled secret societies that supposedly has just come to the "surface." Other people clamor to join.

The "super" suicide society is not so super to Gene, making the name ironic. It meets regularly, because it is a club that Finny is enthusiastic about, but Gene is worried every time he jumps out of the tree (which happens much more than once) because he recognizes the danger. At the same time, he is even more afraid to lose face in front of Finny. The club shows how much Gene is in Finny's grip.

In "A Separate Peace," why is their club named "The Super Suicide Society of the Summer Session"?

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.

In John Knowles's A Separate Peace, what is the significance of the "Super Suicide Society" club name?

To answer a part of your question, yes, author John Knowles intentionally named the club what he named it. He is the author of the book, and writers are typically quite intentional about the names that they give things in their stories.

The more difficult part of the question asks about why Knowles picked that name. Was it meant to be significant for a later event in the book? I think yes. I think that the name of the club directly foreshadows the death of Finny. Finny's death is not a result of suicide, but he still dies. The club's name definitely has an emphasis on death. Had Finny and Gene not been in that tree, they would not have thought about jumping. No jumping, no club. No club, no club title. No club title and no jumping means that Finny never would have broken his leg. That broken leg eventually led to his death, so in a way the Super Suicide Society caused Finny's death. I do believe that Knowles meant for the club name to be intentionally foreboding.