How to Plead for Unbelievers

“Brothers, my heart’s desire and prayer to God for them is that they may be saved.” (Romans 10:1)
Paul prays that God would convert Israel. He prays for her salvation! He does not pray for ineffectual influences, but for effectual influences. And that is how we should pray too.
We should take the new covenant promises of God and plead with God to bring them to pass in our children and our neighbors and on all the mission fields of the world.
God, take out of their flesh the heart of stone and give them a new heart of flesh. (Ezekiel 11:19)
Circumcise their heart so that they love you! (Deuteronomy 30:6)
Father, put your Spirit within them and cause them to walk in your statutes. (Ezekiel 36:27)
Grant them repentance and a knowledge of the truth that they may escape from the snare of the devil. (2 Timothy 2:25–26)
Open their hearts so that they believe the gospel! (Acts 16:14)
When we believe in the sovereignty of God — in the right and power of God to elect and then bring hardened sinners to faith and salvation — then we will be able to pray with no inconsistency, and with the confidence of great biblical promises for the conversion of the lost.
Thus, God has pleasure in this kind of praying because it ascribes to him the right and honor to be the free and sovereign God that he is in election and salvation.
(By John Piper)