According to Josephus, the nobleman was a royal officer or servant (of civil, military, or household service). Consequently, he must have worked in some way for Herod Antipas, tetrarch of Galilee. The name of the nobleman is unknown although some speculate it to be Chuza (Luke 8:3) or Manaen (Acts 13:1). The nobleman's son was sick at Capernaum.
If a son of mine was on his death bed, it would be difficult to remove me from his side. However, if there was any hope for recovery, I would leave his side to investigate and pursue such. No doubt this nobleman loved his son dearly and only left his side to obtain help for him. It appears that the nobleman was not a disciple of Jesus; his faith was not in Christ as the Savior, but in the power he believed the Lord could wield. Since his son was about to die, the nobleman considered Jesus to be his only hope.
Jesus' response in John 4:48 is in the plural form which indicates that He was not just speaking to the nobleman but to all who were present. It was possible to believe in Jesus merely by listening to His words (e.g., 4:42), but most needed a miracle in order to help develop their faith (cf. 20:29-31). Jesus' reply was not a refusal of the nobleman's request. It seems that the Lord was trying to prompt the nobleman to a deeper knowledge of and greater appreciation for Him and His work. However, the nobleman was entirely focused on his son's life at that moment and believed that there was no time to lose!
In 4:49, he plead with Jesus to "come down" before his child died. The nobleman incorrectly believed two things: (1) that Jesus' physical presence was necessary to heal his son and (2) that His powers were only effective on the living (cf. 11:21,39).
Jesus enlarged the nobleman's conception of His divine power by showing him that His words would take effect regardless of distance. Evidently He spoke with such confidence and authority that this father believed and went on his way relieved.
As he journeyed back to Capernaum, he met his servants along the way and they informed him, "Your son lives!" (4:51). He then asked his servants when his son began to get better. The Greek text shows that he had not expected a sudden and complete healing, but the reply of the servants shows that he was mistaken.
The growth in the faith of the nobleman is clearly seen in 4:53. Initially, he had believed in the power of Jesus' presence. Next, he believed in the power of Jesus' words. Finally, he believed in Jesus in a more general sense, and his household shared his belief. This is the first mention of a believing household in the New Testament.
John mentions in 4:54 that this was Jesus' "second sign." This must refer to His second miracle performed at Cana (in addition to turning water into wine; 2:1ff). Jesus had certainly performed other miracles by this time (cf. 2:23).