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)
Here's 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. That's the magnitude of what Jesus invites us into when He calls us to pray in His name.