What does it mean to pray in Jesus' name?
Here's Jesus Himself using the phrase 'in my name':
Many will come in my name, saying 'I am He' and will lead many astray. (Mark 13:6)
This is a very strong understanding of 'in my name'. Here to act "in Jesus' name" is to act as Jesus and to appear to others as Jesus.
Well now - Jesus Himself asks you to pray to the Father 'in His name' (John 14:13,14; 15:16; 16:23). So when you come to the Father, come as Jesus. Come as son, come as christ (anointed one). Call on God Most High with Jesus' own cry - 'Abba, Father' (Mark 14:36 <=> Gal 4:6). By the Spirit, you are so identified with the Son in prayer that you pray as Jesus. And the Father hears you as His very own Beloved.
Jesus does not point the way to prayer, nor simply blaze a trail and ask you to follow along. He incorporates you into Himself in His own self-offering to the Father. As you pray you are not outside the Trinity. But neither are you a fourth member of the Trinity. You are in Christ, filled with the Spirit of adoption, calling on your Abba, Father.
That's prayer "in Jesus' name." But of course we do all of life - whether in word or deed - "in the name of the Lord Jesus" (Colossians 3:17). We don't only approach the Father "clothed in Christ", we approach the world "clothed in Christ." That's our true identity. More true than any phony masks we wear.
I say this stuff with glib assurance but as I write I realise I have no idea of the magnitude of Christ's mediation both to God but also the world!