What does it mean to believe on Jesus' name?
The Name of Jesus represents His divine authority as Creator and Savior.
Be it known unto you all, and to all the people of Israel, that by the name of Jesus Christ of Nazareth, whom ye crucified, whom God raised from the dead, even by Him doth this man stand here before you whole... Neither is there salvation in any other: for there is none other name under Heaven given among men, whereby we must be saved. (Acts 4:10,12)
The name itself is YESHUA, or YAHSHUA (YAH + SHUA) which means
GOD IS [OUR] SALVATION. And only God Himself could be our salvation, and only the blood of God Himself could wash away our sins. So Jesus is in fact the Salvation of God. That is what Simeon said about the child Jesus -- MINE EYES HAVE SEEN THY SALVATION. Thus salvation is in fact a divine Person.
To believe on Him is to believe the Gospel. Which is to believe that:
(1) He is God
(EXCEPT YE BELIEVE THAT I AM, YE SHALL DIE IN YOUR SINS),
(2) He is the only one who obeyed the Law of God perfectly,
(3) He is the only one who could become the Lamb of God, who would take away the sin of the world,
(4) His death, burial, and resurrection are sufficient for the salvation of the whole world, and
(5) you can only be saved by repenting and believing this with all your heart, and then calling upon Him and receiving Him as your own personal Lord and Savior.
But salvation (justification) is purely by the grace of God, through faith in Christ and His finished work of redemption. Thus we receive the gift of eternal life when we receive the gift of the Holy Spirit, and the gift of Christ Himself -- CHRIST IN YOU THE HOPE OF GLORY. When a sinner believes on the Savior, God imputes (or reckons) the righteousness of Christ to his or her account. Therefore we are deemed to be as perfect as Christ.
We cannot add one iota to that, and not one of our righteous works can count for our salvation. Christian baptism by immersion adds absolutely nothing to our salvation, but demonstrates that we are starting on the road to sanctification -- RAISED TO WALK IN NEWNESS OF LIFE. That is the first step of obedience to Christ and His commandments. Which also means the forsaking of all sins and idolatry, and perfect obedience and surrender to God and Christ. Both John the Baptist, Christ, and all the apostles preached REPENTANCE FOR THE REMISSION OF SINS.
ALL OF THIS IS INCLUDED IN BELIEVING ON THE NAME OF THE LORD JESUS CHRIST.