The first instance I have found recorded in the Bible of one person praying for the healing of another is found in Genesis 20.
God tells Abimelech in a dream that he should return Abraham's wife to him and that Abraham will pray for him, God's curse against him will be removed and he and his family will live.
Gen 20:7 "Now return the man's wife, for he is a prophet, and he will pray for you and you will live. But if you do not return her, know that you will certainly die, you and all who are yours.
Gen 20:17 Then Abraham prayed to God, and God healed Abimelech, his wife, and his female slaves so that they could bear children, Gen 20:18 for the LORD had completely closed all the wombs in Abimelech's household on account of Sarah, Abraham's wife."
Our Father has long used the prayers of His children to affect the lives of others in this world.