The traditional view is that the ‘Beloved Disciple’ of the Fourth Gospel is the John, the son of Zebedee, who also wrote (or at least was a major source for) that Gospel.

At first glance, the theory that the Beloved Disciple (BD) was Lazarus seems far-fetched.

Oscar Cullmann thought it possible that Lazarus was the Beloved Disciple and the author of the Fourth Gospel.  Cullmann notes:

(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?

The following, more general, points are also to be noted concerning the author of the Fourth Gospel:

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).

(Source)

Ben Witherington III, in this lecture and in a forthcoming book (Unmasking the Beloved Disciple: The Startling Case for Lazarus as an Eyewitness in the Composition of John’s Gospel, Zondervan, 2027), thinks that this theory makes good sense.

Among other things, this identification clears up a number of puzzles within the Passion narrative of Fourth Gospel itself:

1. This would explain why the BD had ready access to the house of the High Priest.  A Galilean fisherman could harly have had such access.  It is clear from Jn 11:36-47 that some of the Jewish officials who reported to the high priest had known Lazarus, and had attended his mourning in Bethany.  This relationship could have given him access to Caiaphas’ house.

2) This would explain why the scene in the Garden of Gethsemane is omitted in this Gospel.  Peter, James and John were present but the Beloved Disciple (if Lazarus) was not.

3) If the Beloved Disciple took Jesus’ mother ‘unto his own’ home (19:27) this implies a locale much nearer that Galilee.  Indeed, we find the BD in Jerusalem in Jn 20 and Mary is still there in after the crucifixion and resurrection of her son (Acts 1:14).

4) In Jn 20 the Beloved Disciple arrives at the tomb of Jesus before Peter.  This may be because, coming from the Jerusalem area, he knows the locale.  There are, indeed, other indications that he has spent much time in and around Jerusalem, including his knowledge of Joseph of Arimathea and Nicodemus.

5) The Synoptics tell us that all Twelve deserted Jesus from the time he was taken away for execution.  This is not contradicted by Jn 19:26 (“Here is your son…here is your mother”) if the disciple in question is Lazarus, and not one of the Twelve.

6) If the Beloved Disciple took Mary into his own home then we can assume that he got the story of the wedding feast at Cana from Mary herself.