I believe the narrator's first name is mentioned only once in the beginning part of the book. This occurs during the summer session in the context of a game of blitzball, when Finny directs, "Here Gene, the ball is of course still yours" (Chapter 3). The reader doesn't really become aware of the narrator's name, however, until the fall session, which Finny is unable to attend because of the accident at the river. The narrator at this time decides to take on the job of managing the crew team, and he is addressed regularly now by his last name, Forrester" (Chapter 6).
It is significant that the narrator's name is rarely mentioned during the idyllic summer session. This was a time of peace and innocence, qualities which are embodied by Phineas. Finny, in his exhuberant obliviousness to convention and the dour realities of the times dominates everything. Gene Forrester and the others exist in his shadow, a situation which in Gene breeds a guilty sense of resentment which results finally in the tragedy of Finny falling from the tree.
When Finny is taken out of the picture, Gene begins to develop an identity of his own, at least for awhile. The use of his name at this point, when the sheltered haven of summer is replaced by the regimented rigors of fall, parallels his assertion of self. "Peace (has) deserted Devon", and Finny is not there; it is time for Gene to make his way in the real world on his own (Chapter 6).