Believe Jesus is the Christ: The emphasis of John’s purpose statement rests on these signs pointing to the reality that Jesus is the Christ! Least we forget the word “Christ” is not a last name but a title that Jesus takes upon himself (4:25-26). To take upon himself the title of Christ means he takes upon himself to fulfill all the the OT had promised would come about through this future King (Son of David)(see Ps. 2, 110). In other words the emphasis is not so much on our belief and not even in our properly identifying Jesus as a king but rather properly identifying Jesus as the King in whom to Scriptures speak of. The king God Himself identified as and gives His power too.
Son of God: This too is apart of John’s purpose and is somewhat a parallel to the title “Christ” but with some added nuances. In the OT the term “Son of God” was sometimes used as an honorific title when speaking of the Davidic King as God (see Is. 9:6- Mighty God; Ps. 45:6 & 89:28 - Highest One; while at the same time referring to the human son of David 2 Sam. 7, and Ps 2:7, 89:27-28, 110:3). So it seems the title “Son of God” for John is a way to refer to Jesus as the human/ Divine King.
By believing you may have Eternal life in His Name: Since the predominate message throughout John’s Gospel is to believers it seems this part of John’s purpose statement is primarily for the person who already believes to keep on believing because by reading John’s book they are once again affirmed in who He is and thus how they owe this king their full faith and allegiance! The Greek word for “believing” is a “present-active-participle”, meaning its an act that keeps on acting and not simply a one time act of faith. Eternal life is primarily speaking to a “quality of life” and not a “duration of life”. The word “Eternal” is really a substitute word for “God”, in other words Eternal life is “God’s type of life” or as John might say it’s the life of Jesus now being lived out in you. Thus this life is “in His Name”. (which is a poetic way of saying this life is found when we live it in Him, through Him, and in His way.