Is there any evidence that John, the disciple of Jesus, was black?

No, there isn’t; but there is evidence that John was a Jew.

• Jesus’ ministry was primarily exclusive to the tribes of Israel (Mat. 15:24). As John and the other disciples were people Jesus came to minister to, John would have been a Jew.
• Jesus probably selected the twelve disciples to represent Israel in some way (cf. Gen. 49:28; Ex. 24:4; Num. 1:44; Deut. 1:23; Josh. 3:12; Ezra 6:17; Ezek. 48:31; 1QM 2:2-3; 1QM 5:2-3; Pesiq. Rab. 4),¹ which would also indicate that John was an Israelite, a Jew.
• Jesus shared his mission (directed towards the Jews) with his twelve disciples (Mat. 10:5-6), indicating that they (including John), involved in Jesus’ Jewish ministry to Jewish people, were Jews.
• It’s possible that John was Jesus’ cousin,² also an indication of his Jewishness.

A reconstruction of the face of a typical first century Galilean Jew has been made using forensic science and may give some indication of what John might have looked like.

Also see ‘What did Jesus look like?‘ and ‘Was Samson a black man, as depicted in the History Channel series called The Bible?‘

Notes
1. Dale C. Allison, Constructing Jesus (London: SPCK, 2010), p. 71 n. 172. Also see Scot McKnight, ‘Jesus and the Twelve’ (12/05/04) on bible.org (accessed 06/08/15)

2. ‘That John’s mother’s name was Salome is an inference from Mk. 16:1 and Mt. 27:56…Salome is usually regarded as the sister of Mary the mother of Jesus, because [of] Jn. 19:25…If this identification is correct, John was a cousin of Jesus on his mother’s side.’ (R. V. G. Tasker, ‘John, The Apostle’ in New Bible Dictionary 2nd edn [Leicester: IVP, 1962], p. 601)