I'm inclined to say "A blessing is a blessing, is a blessing." I am aware that French Canadian families (maybe others too) have had this beautiful tradition. We don't have to restrict this blessing to New Year's Day.
These are God's blessing instructions to Moses: "The Lord bless you and keep you; may the Lord's face shine upon you and be gracious to you; may the Lord's countenance be upon you and give you peace" (Numbers 6:22-27).
Blessings have deep roots in our biblical tradition. In the Old Testament where they are frequent, blessings are a communication of life from God, bringing strength and success.
God blesses each aspect of creation, including human beings, giving each life and the capability to multiply.
The Catholic Catechism says it beautifully: "From the beginning until the end of time, the whole of God's work is a blessing. From the liturgical poem of the first creation to the canticles of the heavenly Jerusalem, the inspired authors proclaim the plan of salvation as one vast divine blessing" (n. 1078).
Only God can really bless and so, humans only bless by asking God to bless that which they desire to be blest. In the Old Testament, blessing appears to be most common when the one blessing has authority over the person to be blest, thus representing God.
Father to son
The father who has been blest by God transmits his own life, strength and authority to his son, often with hands on the head, indicating the flowing of the blessing from one to the other.
Once given, it cannot be taken back as we see in the story of Jacob deceitfully gaining the first-born's blessing from his father. In the Old Testament, God's blessing is called upon by one person to another, irrespective of authority, as well.
In the New Testament, there are also references to blessings. Jesus blesses the food in multiplying the loaves to feed the multitude and at the Last Supper. Mary is called "blessed among women" in Luke (1:28, 42). The Benedictus "Blessed be the Lord, the God of Israel"(Luke 1:67ff), Zechariah's hymn at the circumcision of his son, John the Baptist, is one of blessing and thanksgiving.
Strictly speaking, since blessings communicate life, only persons can receive blessings. Objects which are set aside for liturgical use, as a new altar or church, holy oils, chalices, etc., are consecrated. However, we usually use the term blessings for these also. Blessings show humanity's dependence upon God and so can be given to other objects such as cars, asking God for safe travelling, etc.
Blessings have first place among sacramentals which "derive from the baptismal priesthood: every baptized person is called to be a 'blessing' and to bless" (Catholic Catchecism n. 1669).
Therefore, it is clear, Christians are expected to bless one another since they are blessed by God "with every spiritual blessing" (Ephesians 1:3). We do this consciously and unconsciously. Our "goodbye" comes from "God be with you," a form of blessing. In today's baptismal ceremonies, along with the priest, both mother and father trace the sign of the cross (a form of blessing commonly used) on the infant's forehead. Parents are encouraged to continue blessing their children in this and/or some other manner.
Whose blessing means more: the priest's or the parents'? Blessings come from God and therefore, sincerely requested in faith, all blessings are effective.
God grants grace
God gives grace in superabundance and we can't really say whether God gives more at the request of one person or another. God can't be outdone in gifting creation.
My biblical theology dictionary indicates that, after God, the source of life comes from the parents and, therefore, blessings given by the father (and mother) are the most powerful.
This is not to deny the value of the priestly blessing, particularly in those actions reserved to the ordained, as for example, the sacraments. Given in the name of the Church, priestly blessings call upon the prayers and the faith of the whole Church and therefore, are especially significant.
Whether we are priests by ordination or Baptism, the constant blessing we are to one another in our way of being and in our words shows our gratitude to God for the superabundance we have received and continue to receive, making us truly a part of that "vast divine blessing."