Many people petition God for someone to be saved and can't understand why it hasn't happened yet. We don't have to ask the Lord to save anyone. He isn't willing that anyone should perish. He has already made provision for everyone's salvation.
So what do we pray concerning someone coming to the Lord? Jesus said to pray that the Lord send laborers across their path. They need to hear the Word. We should bind the influence of the god of this world (Satan) who tries to blind them to spiritual truth (II Corinthians 4:4).
God is more motivated to save our loved ones than we are. We don't need to plead with Him, but rather we need to become a channel for Him to flow through to reach that person. We do that by sharing the good news (Gospel) with them and/or praying that others will come across their path who will do the same.
The Gospel is the power of God that releases the effects of salvation in our lives (Romans 1:16). If a person needs healing, it's in the Gospel. If deliverance is needed, it's in the Gospel. Prosperity, answered prayer, joy, peace, love; they are all found through understanding and believing the Gospel.
The Gospel is the good news that although we are sinners and worthy of God's wrath, God, in love, sent His Son to be our substitute, bearing our punishment, so that we could be made completely righteous in His sight. It's based only on our faith in this completed work of Christ and not our own performance. Now that's GOOD NEWS!
