The main evidence against the idea is that John's gospel does not explicitly mention this happening, and that no oral tradition of such a thing has come down from the early church.
In favour of the idea is that it makes sense of several aspects of John's gospel.
The strongest factor, for me, is the emphasis John gives to the fact that he calls himself the disciple whom Jesus loved; but that this phrase is never mentioned until after the Lazarus raising.
But before this, there has already been a similar emphasis that Jesus loved Lazarus (and also loved Lazarus's sisters, Martha and Mary of Bethany - the Mary who anointed Christ and wiped his feet with her hair, and who I believe to be the same person as Mary Magdalen). The Evangelist goes to considerable length to emphasise both of these loves - for Lazarus and John, and no other man is so described.
If (two assumptions joined here...) John had been Lazarus, and Mary Magdalen was the same person as Mary of Bethany - this would explain why they were both together at the crucifixion.
The other thing this helps to explain is the idea that John the disciple would never die, but would tarry until the second coming of Christ. This is discussed between Jesus and Peter, and apparently comes out of the blue; but if John was Lazarus and therefore had been resurrected then it would make sense, would be a reasonable inference, that John was not going to die as other men die.
As I said, it all fits together except that the Lazarus-John change of name isn't mentioned in the way that the Simon-Peter change is mentioned. This does not rule it out - the fact may have been so well known at the time the gospel was written that it could be taken for granted (after all, John's gospel was written in a way that does seem to take a lot of prior knowledge for granted).
Here are the relevant Biblical passages:
Ch 11 Now a certain man was sick, named Lazarus, of Bethany, the town of Mary and her sister Martha. 2 (It was that Mary which anointed the Lord with ointment, and wiped his feet with her hair, whose brother Lazarus was sick.) 3 Therefore his sisters sent unto him, saying, Lord, behold, he whom thou lovest is sick. 4 When Jesus heard that, he said, This sickness is not unto death, but for the glory of God, that the Son of God might be glorified thereby. 5 Now Jesus loved Martha, and her sister, and Lazarus.
Ch 12 Then Jesus six days before the passover came to Bethany, where Lazarus was which had been dead, whom he raised from the dead. 2 There they made him a supper; and Martha served: but Lazarus was one of them that sat at the table with him.
Ch 13 23 Now there was leaning on Jesus’ bosom one of his disciples, whom Jesus loved.
Ch 19 25 Now there stood by the cross of Jesus his mother, and his mother’s sister, Mary the wife of Cleophas, and Mary Magdalene. 26 When Jesus therefore saw his mother, and the disciple standing by, whom he loved…
Ch 21 20 Then Peter, turning about, seeth the disciple whom Jesus loved following; which also leaned on his breast at supper, and said, Lord, which is he that betrayeth thee? 21 Peter seeing him saith to Jesus, Lord, and what shall this man do? 22 Jesus saith unto him, If I will that he tarry till I come, what is that to thee? follow thou me. 23 Then went this saying abroad among the brethren, that that disciple should not die: yet Jesus said not unto him, He shall not die; but, If I will that he tarry till I come, what is that to thee? 24 This is the disciple which testifieth of these things, and wrote these things: and we know that his testimony is true.