This passage occurs in the Narrative Lectionary, Year 4, Lenten cycle, most recently Feb 9, 2014.
Summary: Did the faith produce the healing? Well, the Greek suggests that belief in Jesus' word only comes after the boy has been healed. In this way, the word produced both the healing and the faith. On the otherhand, the father demonstrates his faith by going to Jesus in the first place. I don't think this story solves this age old conundrum. I don't think it is meant to be solved. What I find more interesting is that Jesus never proclaims the son healed; rather he says the son is alive. I think there are many cases where are ministry isn't about offering people the healing they want, but giving them life, amid grief and illness.
Lastly, I cannot figure out why "go down" appears three times in this passage.
βασιλικος ("royal", 4.46) This adjective is related to the word for king. The NET Bible claims this official must have been related to or working for King Herod. The suggests that the person going to Jesus is willing to engage in risky business. While Herod has not set out against Jesus, Jesus has already upset the temple in Jerusalem and shortly will have people coming after him to kill him.
ζη ("live", 4:50, 51, 53, present tense for of ζαω) I don't know how the translators mess this one up. Every single time the verb is in the present tense: You son is alive. There is nothing future about it. Jesus says and it is so. This is really important because it shows that the healing is not based on the faith of the person. When is the healing accomplished. When Jesus says so. Why? This question is not answered here.
ηρωτα ("ask" imperfect form of ερωταω, 4:47) The man continually is asking for Jesus help here; a sign of faith or despiration?
ιασηται ("heal" aorist form of ιαομαι, 4:47) This verb is not as common as I expected in the Gospels, a few times here and there, mostly in Luke and only once in Mark. It comes into English as psychiatry. I think it is a deep question: Is it our mission, or Jesus' mission, to offer healing? Healing is almost always thrust upon Jesus, the only exception being his command that the disciples go and heal (Luke 9.2). Even here it comes after the proclamation of the Word. To put it another way, healing (and the sick) will come with the proclamation of the Word. The words intnetion is not healing as we see it, but life. I think this opens up more doors -- what is living? How can be sick still have life? In fact, the text next says he was healed of all his problems, just that the fever left him, he was better and he was living.
πιστευω ("believe", throughout this section, including 4:50) This verb means trusting. In this case, trust doesn't produce following Jesus, it creates a situation in which someone can walk away from Jesus. Again, most times we think of trust as creating a situation of moving closer to Jesus, but in this case, the faith creates a situation of letting go, letting go of his anxiety about his son, letting go of his need to be next to Jesus.
καταβαινω ("come down," 4:47,49, 51) Okay, I cannot figure it out. This word appears three times in this story. It appears big time in John 6 (the true bread from heaven). And then it stops. It is as if the incarnation reaches its high point in John's Holy Communion story (chapter 6) and then he is done going down. I'd like some more thoughts on this.