I want to try to show that when Jesus says "Hallowed be thy Name" he is asking God to cause his name to be hallowed. He is not explicitly praying that we do something. The Lord's prayer is important and we ought to get it right.
I will concentrate on Matthew 6, but compare Luke 11.
The Lord's Prayer (Matthew 6:9b-10) The "You" Petitions
Our Father in heaven,
(1) Hallowed be your name
hagiastheto to onoma sou
Let be revered the name of you
(2) Your kingdom come
eltheto he basileia sou
Let come the kingdom of you
(3) Your will be done
genetheto to thelema sou
Let be done the will of you
On earth as it is in heaven.
Comments on Structure
After the address "Our Father in the heavens", Jesus asks for three things to happen. The three lines are related grammatically. The beginning of each line starts with a finite verb in the aorist imperative, followed by a noun as subject and the same singular possessive pronoun. There is no "and" connecting the lines (they are asyndeton) which argues that the three lines should be thought together. The following line "On earth as in heaven" probably should go with all three lines and not just the third. Luke's version omits this line but he also omits the third line "your will be done". The argument is that the last two are certainly petitions which would argue that the first one is as well.
Context
Argument 1: John 12:28
The only other like statement in all the Gospels comes from John 12:28 where Jesus prays to the Father: "Father glorify your name" pater, doxason sou to onomo. The similarities between glorify and sanctify or reverence argues that Jesus is asking God to hallow his name in this verse.
Argument 2: Ezekiel 36:23
This is another (IMHO) knock-down passage. "I will sanctify my great name, which has been profaned among the nations, and which you have profaned among them; and the nations shall know that I am the LORD GOD, when through you I display my holiness before their eyes." This passage is an example of God hallowing his own name and so makes sense of Jesus' line "Let be hallowed the name of you."
Argument 3: Sirach 36:4; Qumran's War Scroll; Jewish Qaddish
Other texts also argu that Jesus' first line is a petition.
(1) Sirach 36:4: "just as you sanctified yourself in their eyes by means of us, so in our eyes reveal your glory by means of them." Just like Ezekiel, it is God who does the santifying.
(2) War Scroll 1QM 11:13-15: Here God will earn an eternal name by doing something that shows him to be great and holy in the eyes of the nations.
(3) Qaddish: Contain the twin ideas of God's sanctifying his name and establishing his kingdom. "Magnified and hallowed be his great name in the world... may he cause his kingdom to reign..." Here again, that God causes his kingdom to reign argues that it is he who sanctifies his name.
Theology
What Jesus is asking is that God sanctify his name by doing acts which will cause the nations (and Israel) to hallow his name. But as John Piper has said, since God is already holy, the purpose of the petition is related to having people hallow God's name, so we get back the common (mis)interpreation of the petition. God's hallows his name by having his name hallowed.
