In the passage, 'Name' signifies not so much the utterance of a word, but the invocation of a power. To know someone's name in ancient parlance meant to be able to call upon them. You did not give your name to strangers. What this statement then means is that whatever is requested in his power he will grant. Another passage specifies that these things must be asked in obedience to him, which in other places we find is what reveals those who love God; those who love him keep his commandments. And given that St. John says, 'God is Love', we can then infer that it is the love of God that makes such prayers be fulfilled, for if God is love, then certainly to ask in his power is to ask in the love of God.
Secondly, 'The Name of the Father and of the Son and of the Holy Spirit' is Jesus Christ. But he and the Father and the Spirit are One, and Jesus is the only personal name for God that we have, that I am aware of. However, given what I wrote above, this passage is talking about asking in the power of God, not so much invoking him by a personal name.
Traditionally, most written prayers have either addressed
- The Father
- The Son
- The Holy Spirit, or
- The Trinity in common
Orthodox Christians have a very ancient set of prayers called The Trisagion Prayers (The prayers of the thrice-holy) which address different persons of the Trinity, as well as the Trinity in common;
The Lord's Prayer addresses the Father,
'O Heavenly King' addresses the Holy Spirit,
All Holy Trinity.. addresses all of the three Persons. The Prayer of the Hours (another common prayer) addresses the Son.
The text of the prayers follows to exemplify that Christ's statement has not been interpreted to deter us from praying except only to him.
To the Spirit
O heavenly King, O Comforter, the Spirit of truth, who art in all places and fillest all things; Treasury of good things and Giver of life: Come and dwell in us and cleanse us from every stain, and save our souls, O gracious Lord.
To the Trinity in common
All-holy Trinity, have mercy on us. Lord, cleanse us from our sins. Master, pardon our iniquities. Holy God, visit and heal our infirmities for thy Name's sake.
To the Father
Our Father, who art in heaven, hallowed be thy Name; thy kingdom come; thy will be done on earth, as it is in heaven. Give us this day our daily bread; and forgive us our trespasses, as we forgive those who trespass against us; and lead us not into temptation, but deliver us from evil.
To the Son
O Christ our God, who at all times and in every hour, in heaven and on earth, art worshipped and glorified; who art long-suffering, merciful and compassionate; who lovest the just and showest mercy upon the sinner; who callest all to salvation through the promise of blessings to come; O Lord, in this hour receive our supplications, and direct our lives according to thy commandments. Sanctify our souls, hallow our bodies, correct our thoughts, cleanse our minds; deliver us from all tribulation, evil and distress. Encompass us with thy holy Angels, that guided and guarded by them, we may attain to the unity of the faith and to the knowledge of thine unapproachable glory, for thou art blessed unto ages of ages. Amen.