In "Word Focus", William Sillings writes:

This man was in earnest about his request. He went unto him, and besought him (v. 47). This construction is made of verbs of varying intensity, and when used together they convey how the man approached Jesus. The first, went unto him (apelthen), is an ingressive aorist indicative indicating that he “went off at once” as soon as he heard that Jesus had come to Galilee. The second, besought (erota), is an imperfect active indicative verb which has the sense, “he began to beg and kept it up.”
While the major purpose of this passage seems not to be intended to teach persistence in prayer, it certainly does illustrate it. This man went down to get Jesus to come back home with him, but Jesus refused with the words, “You won’t believe unless you see signs and wonders.” But this man kept up his request, “Sir, come down before my child dies.” This was a dire situation, and there was no hope outside of Jesus’ intervention.
Perhaps many of us would have given up when we were first denied our request by the Lord. But not this man. Something inside him made him keep asking. The reward for such persistence is that he got what he wanted.

Discussion: How can we learn from this man's persistence in seeking Jesus' help?
Source: Miracles of Jesus: Adult Teacher's Insights, page 64.