In The Johannine Epistles, Ruth B. Edwards states, in parentheses, that the phrase, "you will do well" (3 John 6), is "often associated with a request" (Sheffield, England: Sheffield Academic Press, 1996: 23-24).
This request, coming as it does, not from "John" (he does not use his personal name in the letter), but from "the elder," possibly indicating not merely a personal letter but his capacity as spiritual guide, seems to point to the main purpose in writing: to encourage Gaius not to be cowed by Diotrophes' prohibition, but to (continue to) support the preaching of the gospel through hospitality and financial help as they work in his city and travel beyond.
By saying, "you will do well," the apostle gently urges his friend to ignore one who set himself up as an authority and had forbidden what Christ specifically commanded, that the gospel be preached to all. Diotrophes, said John, "loves to be the leader" (v. 9, NLT; Edwards: "is hungry for power"). Gaius should help the preachers in spite of what the Leader had determined.
Unfortunately, Diotrophes' tribe still exists in the church. And John's short letter remains as a call to resist tendencies to subvert the mission by illegitimate power structures in order to continue the work of God in the world.
We will do well to hear John's request.