Thanks for the question.
When praying directly to the Father we invoke the name of the Son, so that makes perfect sense for that context. A different context of prayer, like praying to Jesus, would involve a different take. Furthermore, when we sign ourselves, we are not really praying but blessing. We are imparting a blessing (albeit to ourselves), in virtue of our priestly status as the people who have been brought into the covenant family of God (cf. 1 Peter 2:9). We have been empowered to act in this way, not apart from God, but in virtue of our new relationship with Him. So this sign follows more of the formula of Baptism. When you bless someone else you can likewise bless them in the name of the Holy Trinity, Father, Son and Holy Ghost; you may have seen priests do this when you ask for a blessing. In this way we identify the Godhead in its fullness and are in essence asking each Person of the Trinity to seal the Blessing. Jesus has established a relationship for us with the whole Godhead and this form of blessing is recognition of that, in addition to being a tradition that has deep roots within the Church.
Though we have a particular way of doing the sign, the earliest use of the sign would have had more primitive forms, like the thumb on the forehead or the use of two fingers. At the end of the second century, Tertullian wrote:
"We Christians wear out our foreheads with the sign of the cross."
This indicates that the sign had been in widespread use and no doubt of Apostolic origin.
- So why impart this blessing?
The answer is simple. When we enter the presence of God to speak with him, we need to remind ourselves that we should be holy, humble and ready to listen.
- Remember how the Lord told Moses to remove his shoes when he encountered the burning bush? (Exodus 3:1-6)
He was suggesting that Moses prepare himself for the divine encounter and recognize the holy presence of God. And Isaiah tells us (in this weekend's reading) that when he saw the Lord, he realized he was a man of unclean lips, so an ember was brought to purify him by an angel to touch his lips. (Isaiah 6:6-8) We have been made clean by the waters of Baptism, and the sign of the Cross, in a small way, re-stirs that cleansing water as a type of sacramental. Think of it as holy water without the wetness. We dip our hands in the font to reclaim that purifying power of God. The nice thing about this sign is that we can do it even in the middle of the desert.