What convinces finny that gene didn't mean to hurt him?

I'm not sure that Finny ever thought that Gene didn't mean to do it rather than he truly forgave Gene's indiscretion. It is good that Finny and Gene come to some sort of reconciliation at that point. Finny forgives Gene for the first time, since he only just stopped denying Gene's responsibility. There is closure in Gene's statements about not hurting Finny out of anything conscious or out of hate, and Finny saying that he believes Gene. When Gene said "this is it" about meeting Finny this time, he knew that they would have to close the issue somehow; with that statement, he foreshadowed his and Finny's reconciliation, and also that something big and final, like Finny's death, was about to happen.