υἱός is most common way to call a child a son. In John, this word is used 57 times. However, there is one instance where παῖς is used:
As he was going down, his servants met him and told him that his son was recovering.
(John 4:51) [ESV]ἤδη δὲ αὐτοῦ καταβαίνοντος οἱ δοῦλοι αὐτοῦ ὑπήντησαν αὐτῷ λέγοντες ὅτι ὁ παῖς αὐτοῦ ζῇ
The official, Jesus and John always call the child a υἱός:
So he came again to Cana in Galilee, where he had made the water wine. And at Capernaum there was an official whose son was ill. When this man heard that Jesus had come from Judea to Galilee, he went to him and asked him to come down and heal his son, for he was at the point of death. (4:46-47)
ἦλθεν οὖν πάλιν εἰς τὴν Κανὰ τῆς Γαλιλαίας ὅπου ἐποίησεν τὸ ὕδωρ οἶνον καὶ ἦν τις βασιλικὸς οὗ ὁ υἱὸς ἠσθένει ἐν Καφαρναούμ οὗτος ἀκούσας ὅτι Ἰησοῦς ἥκει ἐκ τῆς Ἰουδαίας εἰς τὴν Γαλιλαίαν ἀπῆλθεν πρὸς αὐτὸν καὶ ἠρώτα ἵνα καταβῇ καὶ ἰάσηται αὐτοῦ τὸν υἱόν ἤμελλεν γὰρ ἀποθνῄσκειν
Jesus said to him, “Go; your son will live.” The man believed the word that Jesus spoke to him and went on his way. (4:50)
λέγει αὐτῷ ὁ Ἰησοῦς πορεύου ὁ υἱός σου ζῇ ἐπίστευσεν ὁ ἄνθρωπος τῷ λόγῳ ὃν εἶπεν αὐτῷ ὁ Ἰησοῦς καὶ ἐπορεύετο
The father knew that was the hour when Jesus had said to him, “Your son will live.” And he himself believed, and all his household. (4:53)
ἔγνω οὖν ὁ πατὴρ ὅτι ἐν ἐκείνῃ τῇ ὥρᾳ ἐν ᾗ εἶπεν αὐτῷ ὁ Ἰησοῦς ὁ υἱός σου ζῇ καὶ ἐπίστευσεν αὐτὸς καὶ ἡ οἰκία αὐτοῦ ὅλη
Why do the servants tell the man his παῖς is better not his υἱός, as the man calls his son?
