One definition of mediate is to: intercede Christians do this for each other all the time.
"1 This, first of all, I ask; that petition, prayer, entreaty and thanksgiving should be offered for all mankind, 2 especially for kings and others in high station, so that we can live a calm and tranquil life, as dutifully and decently as we may.[1] 3 Such prayer is our duty, it is what God, our Saviour, expects of us, 4 since it is his will that all men should be saved, and be led to recognize the truth; 5 there is only one God, and only one mediator between God and men, Jesus Christ, who is a man, like them, 6 and gave himself as a ransom for them all."
Point 1-4 Paul asks us to Mediate (intercede for each other)
Point 5-6 Because Christ is the eternal bond between the Father and Man.
You have to view 1 Tim 5 in light of the previous verses.
1 Timothy 2:1 I exhort therefore, that, first of all, supplications, prayers, intercessions,
and giving of thanks, be made for all men;
2 For kings, and
for all that are in authority; that we may lead a quiet and peaceable life in all godliness and honesty.
3 For this
is good and acceptable in the sight of God our Saviour;
4 Who will have all men to be saved, and to come unto the knowledge of the truth.
5 For
there is one God, and one mediator between God and men, the man Christ Jesus;
Let's break it down:
I exhort you that: supplications, prayer...FOR all men...that we may lead...for this is good...in the sight of God...who will have all men to be saved...for there is one God...
1 Timothy 2:6 Who gave himself a ransom for all, to be testified in due time.
7 Whereunto I am ordained a preacher, and an apostle, (I speak the truth in Christ,
and lie not a teacher of the Gentiles in faith and verity.
8 I will therefore that men pray every where, lifting up holy hands, without wrath and doubting.
...who gave himself...I will that men pray every where.
All prayer is TO God. All prayer is worship. Prayer may be for others. It may be intercession and supplications. Look at the example that James gives about prayer:
James 5:17 Elias was a man subject to like passions as we are, and he prayed earnestly that it might not rain: and it rained not on the earth by the space of three years and six months.
--Just before this James says: "The effectual fervent prayer of a righteous man availeth much." Fervent prayer, even for another, does not come without worship. Take the Lord's Prayer, not to quote and recite mindlessly as many Catholics do when they hurry through their rosaries. But as Jesus said "after this manner," IOW, to use as a model. Approaching God in his holiness first, "Holy is thy name." Spend time worshiping God praising him of his holiness, of who he is and what he means to you.
Isaiah 6:3 And one cried unto another, and said, Holy, holy, holy,
is the LORD of hosts: the whole earth
is full of his glory.
Revelation 5:13 And every creature which is in heaven, and on the earth, and under the earth, and such as are in the sea, and all that are in them, heard I saying, Blessing, and honour, and glory, and power,
be unto him that sitteth upon the throne, and unto the Lamb for ever and ever.
Praise God in his holiness. The Psalms do much of this.
Confess your sins to God, not to a priest. Why?
1John 1:9 commands us to.
Psalm 66:18 tells us that if we have sin in our heart the Lord will not hear our prayers.
The above indicates that if we do not come and worship the Lord first, then it will be useless to even attempt and intercede for others. Prayer involves worship.
Again take the example of Hannah. Hannah wanted one thing. She wanted a son. This was her request. How does she approach the Lord?
1 Samuel 1:10 And she
was in bitterness of soul, and prayed unto the LORD, and wept sore.
11 And she vowed a vow, and said, O LORD of hosts, if thou wilt indeed look on the affliction of thine handmaid, and remember me, and not forget thine handmaid, but wilt give unto thine handmaid a man child, then I will give him unto the LORD all the days of his life, and there shall no rasor come upon his head.
12 And it came to pass, as she continued praying before the LORD, that Eli marked her mouth.
13 Now Hannah, she spake in her heart; only her lips moved, but her voice was not heard:
The ESV translates verse ten this way:
(ESV) She was deeply distressed and prayed to the LORD and wept bitterly.
She was grieved. She wept as she prayed. In verse 13 we see that her prayer was from her heart. Her lips moved but her voice was not heard. Here was a Godly woman who grieved to have a son, and came to the Lord weeping, worshiping, continuing in prayer (vs.12). It was not just a simple request. She remained a long time in prayer. The context, if you read it, is that she prayed day after day, year after year.
Psalms 126:5 They that sow in tears shall reap in joy.
6 He that goes forth weeping, bearing precious seed, shall doubtless come again with rejoicing, bringing his sheaves with him.
--A Christian's work bathed in the prayer of tears will finally bring forth fruit. Prayer is worship. Prayer is work. Prayer is difficult. Prayer is not saying a rosary. Prayer is getting on one's knees and communing with God--worshiping Him, praising Him, interceding for others, but most of all staying in fellowship with Him.