Let me share six ways that I pray for my unsaved family members and friends, but firstly let me share one way that I do NOT pray. I don't pray that God will save them. Let me explain why.
Firstly, by asking God to save them am I asking Him to do something more so they can be saved? God has already done everything needed for my family's salvation. He has paid the price. The sacrifice has been made. 1John 2:2 says that Jesus IS the propitiation (ie. Full satisfaction, appeasement of God for our punishment) of the sins of the whole world. And 2Cor 5:19 says that Jesus HAS reconciled the world to the Father.
Secondly, am I asking for God to cause them to accept His salvation? God will not overwrite their will in this. He has already said in 2Peter 3:9 that it is not His desire for any to perish. If He could force people to be saved - He already would have.
So how can we pray? Here are different ways that I pray that build upon the fact that God has already completed His work of salvation, and that our loved ones need to come to the place where they receive that salvation by faith.
I pray for the Holy Spirit to bring conviction to them. They need to come to the place where they are convinced of their sinfulness and need of forgiveness. This is the work of the Holy Spirit. Jesus said in John 16:8 that when the Spirit comes He will convict the world concerning sin, and righteousness, and judgement. When I was younger (and more foolish!) I took this responsibility upon myself. I would try and bring my friends into a place of conviction using my own pressures and techniques. Discovering that this was the Holy Spirit's role changed my whole approach. Now I share as the Spirit leads, praying that He will do His work of convicting.
I pray according to Hosea 2:6-7 that God will put a "hedge of thorns" around my loved ones so that they will no longer find satisfaction in the things that they have pursued in the past that have dulled their senses towards God. That just like the prodigal son, they will become aware of their circumstances and realise that it would be so much better to return to the Father (Luke 15:17).
I pray according to Matthew 9:38 that God will bring workers to speak to my family members.
I pray that God will "thrust me into His harvest field" so that I can be a worker in the harvest. As I reach out to those around me and lead them to Jesus, I am becoming the answer to prayer for someone else who is praying for their loved ones. As I am obedient this, I believe that I am sowing a seed that empowers my own prayer for God to send workers to my family.
I pray for them using the eye of faith. I see them through the lens of Jer 29:11 and pray for the plans that God has for them. I begin to call them saved, healed and whole. I give praise for how God will use them to advance His kingdom.
I bind the enemy over them, knowing that we do not wrestle against flesh and blood (Eph 6:12) and the strong man must be bound in order to plunder his goods (Mat 12:29). So I resist the devil (James 4:7, 1Peter 5:8-9) and pray against His work of deception (Rev 12:9, 2 Cor 11:14).