Question: "Is there any Biblical proof to 'believe on' Jesus Christ?'"

The key word being "on" is important.

Anyone can believe "in" God and "in" His Son Jesus Christ. Belief "in" is by intellect. Belief "on" is deeper and is a matter of believing in the heart which requires more than intellect. It requires faith. Belief therefore must be exercised in order to have faith. 

Paul and Silas were in prison for preaching the gospel. There was a great earthquake that rattled so hard, it loosed all the doors and the prisoners. The Philippian Jailer who was on guard at the prison was responsible for all the prisoners and it would be dishonorable if any had escaped. In a moment of "assumption" he was ready to fall on his sword thinking that the prisoners had fled or were going to. However, they all stayed in their places. Paul and Silas had been praying and singing while captive and obviously the Jailor had listened in. He is perhaps overwhelmed too that no one even tried to flee. The Jailor was humbled and became concerned about his salvation. He asked Paul and Silas what he had to do to be saved: Ephesians 2:8 "And they said, Believe on the Lord Jesus Christ, and thou shalt be saved, and thy house." 

How is a person "saved?" It is worth repeating: Ephesians 2:8 "For by grace are ye saved through faith; and that not of yourselves: it is the gift of God:" God extends grace to those who believe through faith and by exercising faith (stronger than knowledge) He hands to one who truly believes the "gift of God" - salvation.

Biblical Proof?

The following verses from my King James Bible all state a person is to "believe on" him. Him being the Son of God, Jesus Christ. With one exception, Romans 10:14, although God does say "believe on" in the verse, there is also the "believe in." One must first believe in the fact that Jesus Christ was born and that He exists. Then as that persons exercises their faith in belief, the equation changes - belief becomes faith. Faith is believing without seeing. Hebrews 11:1 "Now faith is the substance of things hoped for, the evidence of things not seen."

Luke penned:
Acts 19:4 "Then said Paul, John verily baptized with the baptism of repentance, saying unto the people, that they should believe on him which should come after him, that is, on Christ Jesus."

Paul penned:

Romans 4:24 "But for us also, to whom it shall be impute, if we believe on him that raised up Jesus our Lord from the dead;"
Romans 10:14 "How then shall they call on him in whom they have not believed? and how shall they believe in him of whom they have not heard? and how shall they hear without a preacher?"
Philippians 1:29 "For unto you it is given in the behalf of Christ, not only to believe on him, but also to suffer for his sake;"
1 Timothy 1:16 "Howbeit for this cause I obtained mercy, that in me first Jesus Christ might shew forth all longsuffering, for a pattern to them which should thereafter believe on him to life everlasting.

John penned:
1 John 3:23 "And this is his commandment, That we should believe on the name of his Son Jesus Christ, and love one another, as he gave us commandment."
1 John 5:13 "These things have I written unto you that believe on the name of the Son of God; that ye may know that ye have eternal life, and that ye may believe on the name of the Son of God.

It is a matter of the heart, our inner beings, that changes knowledge into faith. And, by faith we are saved. And "faith is the substance of things hoped for, the evidence of things not seen."

John recorded these words: John 20:29 "Jesus saith unto him, Thomas, because thou hast seen me, thou hast believed: blessed are they that have not seen, and yet have believed."

Leave you with a thought to think about...Those who were alive and saw Jesus' face and walked with Jesus the Christ are the only ones who have seen the image of God. Oh that I could have seen His face. Oh glory that I have believed by faith and will one day see His face!