God desires to draw us into His own life. Because He is indescribably holy, we are called to be transformed by His grace and become holy.
This is what is meant by the petition “hallowed be Thy Name,” that God’s Name be made holy in us, and in others. As sinners still on the path to sanctification, we pray for God’s saving plan to be accomplished in our lives.
God has entrusted us with a great mystery of intimacy — His revelation of Himself, first to the People of Israel, and ultimately in the Person of Christ.
But how do we respond to this divine gift? If we treasure the holiness of God and strive to live in accord with His grace, we hallow His Name. On the other hand, if we deliberately indulge in unrepentant sin, or disparage sacred things, we besmirch His Name, effectively telling God that we don’t care about Him or His friendship.
That is why the saints have been so zealous for the Holy Name: God’s friends yearn to glorify Him.
For more, see Catechism paragraphs 2803-15.