I have always being taught that the beloved disciple that Jesus referred to in Jn 13:23 is John, son of Zebedee and writer of the Gospels of John, the Johannine epistles and the book of Revelation.
Ben Witherington in a lecture given last year identified the beloved disciple to be…. Lazarus!
It was to Lazarus, his beloved disciple that Jesus entrusted the care of his mother when he hung on the cross.
In addition, Witherington thinks that Lazarus was the son of Simon the Leper and Lazarus himself died of leprosy. When Jesus resurrected Lazarus, he also cured him of his leprosy. That will make Mary, Martha and Lazarus lepers as they are of the household of Simon the Leper. Read more here