In John 17:20, Jesus prayed for the disciples and the lost.
We have lots of power when we "remind" God of His promises. Praying "in his name" means remaining in him and having the Word of God in us. Jesus is the power source, and His Holy Spirit draws people.
8 things to pray for:
1. Cleansing: what we have that they don't (Ephesians 5:26-27)
We are cleansed by hearing the Word. Other things we listen to, read, talk with people about, etc., affect cleansing.
2. Love and unity
I can't speak against people if I don't pray for them.
3. Greater faith (Romans 10:17)
I can't condemn myself or think that God will give up on me because of my little faith. A man actually asked Jesus to help his unbelief. See 1 John 3:21-22 and James 2:5-6)
4. Spreading of the word of God
5. Work of the Holy Spirit (e.g. conviction)
6. Laborers (people to witness)
7. Acts of love (ministering to needs without pay) (John 13:34-35)
8. Christ will be lifted up in all things that we do.