So with this in mind, it struck me that the Our Father is a prayer that could be seen as a string of reminders of the love God has for us and of our love of God and neighbor.
"Our Father who art in heaven": The mere fact that God reveals Himself to us as Father means that He provides for us and also means that He gives Himself to us in an intimate relationship of Love.
"Hallowed be Thy Name": In the Old Testament, the Name of God (YHWH) was held as sacred because its meaning was a reminder to the people of Israel that "I am He Who is in your midst." In other words, for the LORD (the use of all caps in the English translations of the Bible is a means to translate the use of the name of God, YHWH, while still being respectful to the sensitivities of the Jewish people to not pronounce it) to remind His people of His Name after instructing them in His commandments is to place the emphasis on God's gift of Himself to His people first and foremost and to instruct them that the law is to be lived as a response of love to God Who first loved them and reveals Himself to them.
In the book of Malachi, the LORD chastises His people for dispising His Name by not offering the best of sacrifices. In other words, some of the people thought that if they simply offered a sacrifice, this would put them in God's good graces. They would not offer the best out of love for God but simply look to instrumentalize the temple. What's more, the LORD chastises His people in the Old Testament for offering sacrifices without any concern for moral uprightness, and in this way, He says they are desecrating His Name among the nations.
So, to Hallow the Name of God is not only to keep in mind God's first love... the gift of Himself in revealing Himself as He Who is in our midst, but to then act out of this love, ie to love God and one's neighbor and not simply say one loves God.
So by saying "Hallowed be Thy Name", not only are we recalling God's gift of Himself to us, but we are asking also that He give us the grace we need to truly "Hallow" His name by living out the command to love Him and our neighbor as a response to His wonderous and gracious love.