Does the New Testament mention skin color? Yes, I know of one instance: in Acts 13:1, as follows.
Acts 13:1 Ἦσαν δὲ ἐν Ἀντιοχείᾳ κατὰ τὴν οὖσαν ἐκκλησίαν προφῆται καὶ διδάσκαλοι ὅ τε Βαρναβᾶς καὶ Συμεὼν ὁ καλούμενος νίγερ, καὶ Λούκιος ὁ Κυρηναῖος, Μαναήν τε Ἡρῴδου τοῦ τετραάρχου σύντροφος καὶ Σαῦλος.
¹Now among that congregation in Antioch there were prophets and teachers: both Barnabas and Simeon― the one called “the black Simeon”; and also Lucius the Cyrenian; Manaen the foster brother of Herod the Tetrarch; and Saul.

The one called “the black Simeon” was called that so that he not be confused with Simeon the apostle, Peter. For some reason, in Acts 15:14 the apostle Peter is called Simeon rather than the usual spelling Simon.

I think it is unacceptable that most translations do not translate the Greek word Νίγερ, but only transliterate it as Niger. A couple even say it was his last name, “Simeon Niger.” But the lexicons say it is a loanword from Latin which clearly and obviously means “black.”

The black Simeon was a prophet or teacher in the Christian congregation in Antioch, Syria.