"First of all, then, I urge that supplications, prayers, intercessions, and thanksgivings be made for all people, for kings and all who are in high positions, that we may lead a peaceful and quiet life, godly and dignified in every way. This is good, and it is pleasing in the sight of God our Savior, who desires all people to be saved and to come to the knowledge of the truth. For there is one God, and there is one mediator between God and men, the man Christ Jesus, who gave himself as a ransom for all, which is the testimony given at the proper time." - 1 Timothy 2:1-6 ESV
There is so much packed into these few verses. Each word and the voice tense and mood of each verb is so telling. The God of the Bible is a God who desires salvation for all people. This God says that our prayer lives matter and that everyone needs prayer. Peace, quiet, Godliness and dignity are linked to prayer for people and for leaders. Our mediating ministry is linked to Christ's mediating ministry.
I think prayer is so much about awareness. People condemn themselves for their own prayerlessness but that is the opposite of what a prayer is. Condemnation blocks us and makes us double minded. Our awareness of a gracious God must meet the awareness of the soul of another person and only in the living Spirit filled soul of the believer can that intercession happen. Spiritually blind people can't navigate prayer. The words are just an overflow of knowing someone and trusting what we know about God. The words are a hand on the rudder of a sailing ship that is being moved by a great wind.
"Rejoice always, pray without ceasing, give thanks in all circumstances; for this is the will of God in Christ Jesus for you. Do not quench the Spirit." - 1 Thessalonians 5:16-19 ESV
Those four words... supplications (Deēsis - Strong's G1162), prayers (Proseuchē - G4335), intercessions (Enteuxis - G1783), and thanksgivings (Eucharistia - G2169) are great words to study.
In a way they are four aspects of prayer. Seeking, public open air asking, conferencing and thanking from a heart overflow of recognition.
People can pray for us but no one can pray as us. The prayers we pray for a person are special to God. Nothing can take their place. Prayer fills a space. Prayer makes a space.
The picture I get is of blowing up a balloon with one person's name on each balloon. The air is grace thinking and grace asking. Soon the room is filled with balloons and the names are large and easy to read. Today the people we know can go unnoticed by us or we can fill our room with floating representations or their names. We can mention those names to God again and again.
"For I know the plans I have for you, declares the LORD, plans for welfare and not for evil, to give you a future and a hope." - Jeremiah 29:11 ESV
We can't know God's plans for people but we can think the way He thinks and mediate in communion with the great and only true mediator. We can say on Earth what Jesus is saying in Heaven when we think in grace and purpose to notice people and actively believe that God has a desire to save and impart hope in each and every circumstance.