What does it mean to pray in Jesus name? We learn from reading John 14:13 that whatever we ask in Jesus name, this He will do that the Father may be glorified in the Son. Looking at the first Christian martyr Stephen, we see the connection between a true Christian who held a relationship with the Son as it was meant to be.
Although the apostles went about healing people and performing miracles stated as "in Jesus name", when Stephen was being stoned, he had a vision of Christ standing next to the Father and he addressed his prayer in Jesus name directly to Him. Where one translation of the bible totally botches this meaning by replacing God's title Lord with the Father of creation's name everywhere it is not, all others show the truth of this faith.
When they heard these things they were cut to the heart, and they gnashed at him with their teeth. But he, being full of the Holy Spirit, gazed into heaven and saw the glory of God, and Jesus standing at the right hand of God, and said, "Look! I see the heavens opened and the Son of Man standing at the right hand of God!"
Then they cried out with a loud voice, stopped their ears, and ran at him with one accord; and they cast him out of the city and stoned him. And the witnesses laid down their clothes at the feet of a young man named Saul. And they stoned Stephen as he was calling on God and saying “Lord Jesus, receive my spirit." Then he knelt down and cried out with a loud voice, "Lord, do not charge them with this sin." And when he had said this, he fell asleep.
-Acts 7:54-60
If you have ever called on God in a crushed spirit, or in desperation, it is easy to relate to this way of praying.