Why does John not mention himself by name in the Gospel of John?

The short answer is that John does not mention himself by name in the Gospel of John because John did not write that Gospel. It was written anonymously and only attributed to John by the Church Fathers later in the second century, when they were attempting to decide who probably wrote each of the New Testament gospels.

A slightly fuller anwer is that the Church Fathers, puzzled at the lack of evidence as to who wrote this Gospel, saw that a 'disciple whom Jesus loved' was a key character in this Gospel. They decided that this disciple must be the author, who must have just been too modest to use his own name. They then noted that the apostle John was not mentioned any where in this Gospel, and decided that John was the missing disciple. Thus, on supposition and quite limited evidence, John became the author of the fourth gospel.