They "... explained to him the way of God more accurately ..."
"... if he found any who were of the Way,
whether men or women,
he might bring them bound to Jerusalem ..."
How did this name, The Way, come about? The obvious answer to Believers is "Because Jesus is The Way" (John 14'6). He is the One who preached, taught and lived God's Kingdom life.
He preached of the reality of the Kingdom of God and He also demonstrated it. After the Father had sent the Holy Spirit the new Believers began to do the same and were described accordingly.
They were like their Lord, expressing the Kingdom to the world (and to the powers of darkness!). Also, like Jesus, their desire was to glorify their heavenly Father.
The Good News spread out from Jerusalem. And these men and women who had begun following Jesus were described as people "of the Way".
As far as the Kingdom of God on earth is concerned, He showed Himself as the example of being its first citizen.