Though each person of the Trinity are equal in their nature and being, they are not equal in their relation to each other in their roles. As a father has authority over a son, or a husband has authority over his wife, or a parent over a child, each of these remain equally human. Equally dignified. The Son has always been subordinate to the Father, and the Spirit subordinate to both of them. Submission is thus a noble thing shown within the Trinity, which helps us creatures understand it better in our own spheres (And thus crumbles many of the arguments for egalitarianism and feminist rhetoric).
The order of the names may denote that sense of authority. We see that in ancient writings and elsewhere in the Bible. We say ''The Father, the Son, and the Holy Spirit'' in that manner (As like in baptism) because that's the command. It would not be sinful to talk of God in a different order, though it would be in error if you were meaning to call them all equal in roles or alter those roles in any way.
And as for being ''in the name,'' that denotes the authority and character of that person. That is why Christians commonly pray in the name of Christ. It's by His merit we can pray. Also, Jesus commands that His people pray in His name, but this also means praying in His will, character, etc (Thus also crumbles a lot of rhetoric from the prosperity gospel camps...).
''Haste thee on from grace to glory,
Armed by faith, and winged by prayer.
Heavens eternal days before thee,
Gods own hand shall guide us there.
Soon shall close thy earthly mission,
Soon shall pass thy pilgrim days,
Hope shall change to glad fruition,
Faith to sight, and prayer to praise.''