What does it take to convince an unbeliever to accept Jesus Christ?

The short answer to this question is that it takes ZERO convincing! It is not our role to convince a person. As a matter of fact, how can you convince an unbeliever of spiritual issues. You can talk until you are blue in the face, but when you try to talk, convince, or argue someone into faith, they may concede to the strength of your argument, but faith was not produced. No person arrives at the point of genuine "saving faith" by the efforts of another, or of their own mental ascent. It is the result of God's work/grace in our spirit by His Holy Spirit conviction. 

Sharing the gospel is like sowing seeds. That's all you can do. Sow the seed and pray that it falls into fertile soil. It is the Holy Spirit that uses the gospel to convict an unbeliever of sin, righteousness, and judgement. The Holy Spirit will produce moments of truth in that person's heart, moments where they will be confronted with a decision to believe. It is at this point that they can choose life and get onto that narrow path or continue on the wide road of destruction. Have a look at how the Bible highlights these moments of truth,

Acts 2:37 Now when they heard this, they were cut to the heart, and said to Peter and the rest of the apostles, "Men and brethren, what shall we do?" 38 Then Peter said to them, "Repent, and let every one of you be baptized in the name of Jesus Christ for the remission of sins; and you shall receive the gift of the Holy Spirit". 

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? 15 And how shall they preach unless they are sent? As it is written: "How beautiful are the feet of those who preach the gospel of peace, who bring glad tidings of good things!"> 

God prepares our hearts for the Gospel; God alone draws and convicts us.

Salvation "by grace through faith" requires God's enablement – "it is the gift of God" (Ephesians 2:9). An unsaved person cannot be convinced of spiritual truth by the 'human' messenger. Neither can that unsaved person pump up (or create in themselves) genuine faith to believe; Why not? Because they are spiritually dead. (1 Corinthians 2:14; 2 Corinthians 4:3; 1 Corinthians 1:18; Ephesians 2:1).

Paul confirms this when he wrote,

Galatians 1:15-16 But WHEN IT PLEASED GOD, who separated me from my mother's womb, and called me by his grace, 16 TO REVEAL HIS SON IN ME...

John provides additional insights,

John 16:7-11 Nevertheless I tell you the truth. It is to your advantage that I go away; for if I do not go away, the Helper will not come to you; but if I depart, I will send Him to you. 8 And when He has come, He [not the pastor or evangelist] will convict the world of sin, and of righteousness, and of judgment: 9 of sin, because they do not believe in Me; 10 of righteousness, because I go to My Father and you see Me no more; 11 of judgment, because the ruler of this world is judged.

God's grace at work

After the seeds of the gospel are sown, and watered in prayer, through God's grace, the Holy Spirit draws an unbeliever into a godly conviction where their response is heartfelt and real, where there is a needful desire or wanting to draw near to God. More examples in the Bible of God doing the work to draw people to Himself through the truth and power of the gospel message,

Acts 16:14 And a certain woman named Lydia, from the city of Thyatira, a seller of purple fabrics, a worshiper of God, was listening; and the Lord opened her heart to respond to the things spoken by Paul.

1 Corinthians 1:30 ... but of Him you are in Christ Jesus, who became for us wisdom from God—and righteousness and sanctification and redemption— 31 that, as it is written, "He who glories, let him glory in the Lord."

Ephesians 1:18 ...the eyes of your understanding being enlightened; that you may know what is the hope of His calling, what are the riches of the glory of His inheritance in the saints,

Titus 3:5 not by works of righteousness which we have done, but according to His mercy He saved us, through the washing of regeneration and renewing of the Holy Spirit.

Hebrews 12:2 Fixing our eyes on Jesus, the author and perfecter of faith, who for the joy set before Him endured the cross, despising the shame, and has sat down at the right hand of the throne of God.

So, in summary, the 'human' messenger sows the message of the gospel. He does not press for a positive result, he does not press him for an answer, he does not try to convince him. Why! Because the messenger cannot save the unbeliever! The conviction of the unbeliever is the work of the Spirit. The decision the unbeliever makes is between himself and God and would be in response to the faith that God puts on his heart at that moment when his heart is contrite and receptive. Again, this is a moment entirely between himself and God and not a man-made outcome. All that messenger can do is pray for that moment of grace to come into the heart of the unbeliever.