Good question, and quite relevant, hermeneutically. My answer to your question is no. When Jesus commanded "the eleven" to baptize disciples "in the name of the Father, and of the Son, and of the Holy Spirit," He was not speaking of names, literally.
Does God the Holy Spirit have a "real" name in the same way Jesus does? Well, we do have several biblical titles or descriptions of the Holy Spirit's role within the Godhead. Perhaps the most encouraging title of all to us is Comforter, or in the Greek, paracletos, which means primarily one who is called alongside another person, but also one who provides comfort and counsel (see this web site; also John 14:16, 26; 15:26; and 16:7).
A better question might be, "Do the Holy Spirit and the Son of God have a name in common with the Father?" And the answer is yes. Recall the experience of Moses, in Exodus chapter 3:
"Then Moses said to God, 'Behold, I am going to the sons of Israel, and I will say to them, The God of your fathers has sent me to you.' Now they may say to me, 'What is His name?' What shall I say to them?' God said to Moses, 'I AM WHO I AM'; and He said, 'Thus you shall say to the sons of Israel, 'I AM' has sent you. . . . This is My name forever, and this is My memorial-name to all generations" (vv.13-14; v.15b).
God the Father, God the Son, and God the Holy Spirit have many different titles, each of which reflects a certain role or aspect of His being. The Godhead, however, has only one name, and that is YHWH (or YHVH), the four letters of which have been called the Tetragrammaton, which is the Hebrew word for God. That sacred Name consists of the four Hebrew letters yod, he, vav, and he, transliterated consonantally usually as YHVH. For centuries before Christ, pious Jews would not even pronounce The Name out of reverence for God, whom they spoke of as Adonai (Lord) or Elohim (Almighty God), or as any one of numerous compound names of God (e.g., YHWH-Rapha, YHWH-Jireh, and YHWH-Sabbaoth, which mean respectively, God is Healer, God is Provider, and Lord of Hosts).
Historically, the phrase "in the name of" was a common locution in Jesus' day, and in the realm of politics and government, for example, to be summoned "in the name of the Emperor" would strike fear in the hearts of most people. They might wonder to themselves, "Oh no, what did I do wrong? Did I displease Caesar? Why am I being summoned before His Majesty?" The apostle Paul, on at least one occasion and as a Roman citizen appealed to the name of Caesar (Acts 25:11-12; 25:21; 26:32; 27:24; and 28:19). Today we might liken Caesar to our Supreme Court, which in America is the court of final appeals. No court is higher.
The phrase "in the name of Caesar," then, struck fear in the hearts of people because the Emperor in that day had virtually unlimited authority and power. The life of each of his subjects was in his hand, as it were, and with one word from his lips their lives could be snuffed out. To bring this scenario up to date, you might think of being summoned today by a governmental body. A letter arrives at your home, the first paragraph of which says,
"Your presence is requested by the Internal Revenue Service for an audit of your tax returns for the years . . .."
Now that would strike fear in anyone's heart! The reason, in part, is that the very name of that government agency wields tremendous power over the lives of American citizens, particularly when it thinks a citizen has not been paying his or her fair share to the government's coffers!
This notion of tremendous power or absolute authority, then, is at the very heart of Jesus' words, "In my name."
We know that many Evangelical Christians are in the habit of ending their prayers (at least in public) with the phrase
"In Jesus' name."
There is nothing inherently wrong in doing so. After all, Jesus Himself told us to pray in that way:
John 14:13 NIV - And I will do whatever you ask in my name, so that the Son may bring glory to the Father.
John 14:14 NIV - You may ask me for anything in my name, and I will do it.
John 14:26 NIV - But the Counselor, the Holy Spirit, whom the Father will send in my name, will teach you all things and will remind you of everything I have said to you.
John 15:16 NIV - You did not choose me, but I chose you and appointed you to go and bear fruit--fruit that will last. Then the Father will give you whatever you ask in my name.
John 16:23 NIV - In that day you will no longer ask me anything. I tell you the truth, my Father will give you whatever you ask in my name.
John 16:24 NIV - Until now you have not asked for anything in my name. Ask and you will receive, and your joy will be complete.
John 16:26 NIV - In that day you will ask in my name. I am not saying that I will ask the Father on your behalf.
Nevertheless, Jesus did not teach His followers to use the phrase in my name as a magical mantra! Back in the day (well, back in my day), soul singer Janice Joplin sang plaintively,
"Lord, won't you buy me a Mercedes Benz?"
We do Jesus a disservice when we treat Him as a magic genie who is always there to grant us whatever our hearts desire.
Jesus was not encouraging us in His teaching on prayer simply to append the phrase "in my name" to every prayer as a sort of verbal talisman which gets results (see this website). Go back to the Roman Emperor's summons. Suppose you as a parent were to use the name of the Emperor (e.g., Caesar Augustus) without his explicit permission.
"In the name of Caesar Augustus, I demand that you release my son from prison this instant!"
The demand is only as good as the authority of the person in whose name the demand is made, not to mention the legitimacy with which the name is used. A person who forges an official looking release document and gets caught doing so will soon find himself in jail, and tout de suite!
In the same manner, our prayers "in Jesus' name" appeal to the authority and power of the King of kings and the Lord of lords. When we invoke His name, we are not only invoking His literal name, Jesus (as precious as it is), but perhaps more importantly we are invoking the name of a person who said of Himself,
"All authority has been given to Me in heaven and on earth. Go therefore . . ." (Matthew 28:18b-19a, my emphasis).
It is in our going and making disciples of all people-groups that we are to invoke the authority of the one who commissioned us to "go and make disciples." In other words, we invoke His name when our desire above all other desires is to do His will, to do what delights Him, and that is to give more and more of the people whom He loves
"the right to be called children of God," even to "as many as [simply] receive Him and believe in His name" (John 1:12).
In addition to the authority of Jesus, in our praying we also need to consider the glory of God (see John 14:13, above). A request in Jesus' name is our way of saying we want Jesus to grant our request so as to glorify His--and our--Father in heaven. Selfish and self-centered prayers have a way of bouncing off the ceiling. Prayers which reflect our sincere desire to enhance God's reputation in the world are a sweet-smelling savor to God, and He delights in answering those prayers affirmatively.
Authority and power, God's glory, and God's will are perhaps the three most important keys of the kingdom. Only when they are in perfect alignment can we be assured of God's hearing our requests and granting them in His Son's name.