Who was the author of the Fourth Gospel? Tradition has identified John the Apostle as the author of it (or a circle of his disciples, i.e. the Johannine Community), and some early manuscripts have added the heading "The Gospel According to John" to the beginning or end of the Fourth Gospel, and this gospel is commonly known as the "Gospel of John". However, the internal evidence of this gospel points to another author than the Apostle John. The Fourth Gospel is internally anonymous, and its author cryptically refers to his or her self as "the disciple whom Jesus loved" and there is good reasons to believe that this Beloved Disciple was not the Apostle John, but someone else.

Oscar Cullmann argues in his The New Testament: an Introduction for the General Reader that the true author of the Fourth Gospel was Lazarus of Bethany, and provides the following compelling (yet speculative) argument that Lazarus was the Beloved Disciple who wrote the Fourth Gospel.   

We would, on the other hand, be tempted to propose a rather fascinating hypothesis, although it is not sustained by indisputable arguments: the beloved disciple == Lazarus who was raised from the dead. Here are the factors that can be brought up in favor of this hypothesis:

(1) The fourth Gospel is the only one that mentions Lazarus, and it places him among Jesus' close companions. It is also the only one that mentions a beloved disciple who, although he is one of Jesus' intimate friends, is not necessarily part of the group of the twelve apostles. 

(2) Lazarus is the only person of whom it is also said that Jesus loved him (cf. John 11:3, 5, 35-36). 

(3) When John 21 informs us the rumor going around that the beloved disciple would not die, might we not think of Lazarus whom Jesus had resurrected from the dead? 

But it must not be forgotten that this identification is entirely hypothetical. We must resign ourselves to not knowing with certainty the name of the beloved disciple. However we can say this about him:

a. He comes from a different theological world than that of the other Evangelists, perhaps from the Hellenists of Palestine or Syria [. . .]

b. He is not necessarily part of the group of the Twelve who, as such, do not have any role in this Gospel, whereas it mentions other intimate disciples of Jesus;

c. He does not seem to belong to the same social milieu as the other disciples (he was an acquaintance of the High Priest, cf. John 18:15-16);

d. He is perhaps from Jerusalem (historically he is well-informed concerning Jerusalem traditions).

Oscar Cullmann's speculation that Lazarus was the Beloved Disciple that wrote the Fourth Gospel, is comparable to other speculations such as whether Pricilla or Apollos wrote the anonymous Epistle to the Hebrews, etc. I highly recommend reading this short book by Cullmann, for many other fascinating insights into the New Testament. 

Sources:

1. Oscar Cullmann, The New Testament: an Introduction for the General Reader, tr. Dennis Pardee, Westminster Press: Philadelphia, 1966, p. 53 [bible citations format modified to work with reftagger plugin]