What, therefore, is the meaning of “hallowed be your name”?…

When it is our settled conviction and belief that he who by nature is God over all is Holy of the Holies, we confess his glory and supreme majesty. We then receive his fear into our mind and lead upright and blameless lives. By this we become holy ourselves, and we may be able to be near unto the holy God. … The prayer is, therefore, “May your name be kept holy in us, in our minds and wills.” This is the significance of the word hallowed. If a person says, “Our Father, hallowed by your name,” he is not requesting any addition to be made to God’s holiness. He rather asks that he may possess such a mind and faith to feel that his name is honorable and holy. The act is the source of life and the cause of every blessing. How must being this influenced by God be worthy of the highest estimation and useful for the salvation of the soul?

Cyril of Alexandria, Commentary on Luke, Homily 72, quoted in Arthur A Just (ed), Luke, Ancient Christian Commentary on Scripture, New Testament III, (Intervarsity Press, 2003) 186.