In English the word "name" means:

a word or a combination of words by which a person, place, or thing, a body or class, or any object of thought is designated, called, or known. -- dictionary.com

However, in both Hebrew and Greek the meaning of name is much more than a designation or identifier. For example the following verse make no sense if "name" means a way to identify someone:

His eyes are like a flame of fire, and on his head are many diadems, and he has a name written that no one knows but himself. -- Revelation 19:12 (ESV)

What does it help to have a name that only he himself knows?

The meaning of the word "name" in Hebrew שֵׁם ("shem"):

• name (to identify), for example Genesis 4:17 (name of person), Genesis 2:11 (name of river), Genesis 2:19-20 (name of beasts), Genesis 26:33 (name of city), Ezekiel 24:2 (name of day)
• character, for example Proverbs 21:24
• ownership, for example 2 Samuel 12:28
• reputation, for example Genesis 12:2, 2 Samuel 7:9, Jeremiah 32:20, Nehemiah 9:10
• designation, for example Genesis 4:26

The same word "shem" is also translated as

• "fame" in Zephaniah 3:19
• "renown" in Ezekiel 39:13
• "byword" or "infamous" or "subject of gossip" or "notorious" in Ezekiel 22:5, Ezekiel 23:10

The meaning of the word "name" in Greek ὄνομα ("onoma"):

Original Word     : ὄνομα, ατος, τό
Part of Speech    : Noun, Neuter
Transliteration   : onoma
Phonetic Spelling : (on'-om-ah)
Definition        : a name, authority, cause

• name (to identify), for example Mark 3:16 (name of a person)
• authority
  • "as a rank", for example Matthew 10:41
  • "on behalf of", for example Acts 10:48, Mark 9:38, Acts 1:15
  • "in obedience of", for John 5:43, Matthew 28:19, Acts 8:16
• reputation, for example Luke 1:49, Matthew 6:9

"Name" as a magic word

Some bad translations may read:

At that time you won’t need to ask me for anything. I tell you the truth, you will ask the Father directly, and he will grant your request because you use my name... -- John 16:23 (NLT)

... as a magic word?

Jesus will never say that. There are no magic names in the Bible. A more accurate translation reads:

And in that day ye shall ask me nothing. Verily, verily, I say unto you, Whatsoever ye shall ask the Father in my name, He will give it you. -- John 16:23 (KJV)

"Name" as an authority

The term "in the name of ..." means authority. This is evident in the following scriptures:

And when they had set them in the midst, they inquired,

“By what power or by what name did you do this?”

Then Peter, filled with the Holy Spirit, said to them,

"Rulers of the people and elders, if we are being examined today concerning a good deed done to a crippled man, by what means this man has been healed, let it be known to all of you and to all the people of Israel that by the name of Jesus Christ of Nazareth, whom you crucified, whom God raised from the dead — by him this man is standing before you well..."

-- Acts 4:7–10 (ESV)

Peter said, "...In the name of Jesus Christ the Nazarene – walk!" -- Acts 3:6

It is not Peter's "magic", he commanded under the authority of Jesus Christ that healing should take place. It was God's power that did the healing. Peter was only the agent that acted, in this case commanded, on behalf of God.

But Paul was greatly annoyed, and turned and said to the spirit, "I command you in the name of Jesus Christ to come out of her!" -- Acts 16:18

In this case Paul commanded a spirit to come out on behalf of God, in His authority.

And the crowds that went before him and that followed him were shouting, “Hosanna to the Son of David! Blessed is he who comes in the name of the Lord! Hosanna in the highest!” -- Matthew 21:9 (ESV)

The crowd would not have reacted if "the Son of David" was coming because his name was "Jesus" because "Jesus" was a common Hebrew name and many other Jews were also called "Jesus". Zechariah 9:9 's prophesied this event. This prophecy was given by YHVH (pronounced "Yahweh" or "Jehovah", but translated as "Lord") as seen in Zechariah 9:1 which makes his event significant and gives Jesus the right to be treated as a king. Therefore, the crowds recognized that Jesus was coming in the authority of YHVH (which is unfortunately translated as "Lord").

“Now we command you, brethren, in the name of our Lord Jesus Christ, that you keep aloof from every brother who leads an unruly life and not according to the tradition which you received from us.” -- 2 Thessalonians 3:6 (ESV)

We also find this in the Old Testament. For example Moses' problem was not that he did not know how to identify God. Moses already identified God's name as YHVH. Moses' problem was that if he returns, as a coward exile shepherd with no rights or authority in Egypt, he had no chance of success. However, God told him that He as the eternal God of their forefathers is sending Moses with His "name" (authority) to deliver a message from Him.

Then Moses said to God, “Indeed, when I come to the children of Israel and say to them, ‘The God of your fathers has sent me to you,’ and they say to me, ‘What is His name?’ what shall I say to them?”
And God said to Moses, “I AM WHO I AM.”
And He said, “Thus you shall say to the children of Israel, I AM has sent me to you.’ ”
Moreover God said to Moses, “Thus you shall say to the children of Israel: ‘The LORD (YHVH) God of your fathers, the God of Abraham, the God of Isaac, and the God of Jacob, has sent me to you. This is My name forever, and this is My memorial to all generations.’ Go and gather the elders of Israel together, and say to them, ‘The LORD God of your fathers, the God of Abraham, of Isaac, and of Jacob, appeared to me, saying...” -- Exodus 3:13-16 (NKJV)

Isaiah did not prophesy that God would change everyone's names to His name, but rather that God give a command or authority to gather the ignorant (blind/death) people:

Everyone who is called by My name,
Whom I have created for My glory;
I have formed him, yes, I have made him.
Bring out the blind people who have eyes, and the deaf who have ears.

-- Isaiah 43:7-8 (NKJV)

Commandment

You shall not take the name of the LORD your God in vain, for the LORD will not hold him guiltless who takes his name in vain. -- Exodus 20:7 (ESV)

According to the Interlinear Bible, the direct translation is:

Yahweh will hold [him] guiltless not for in (vain / nothing / worthless / falsehood) your God of Yahweh the (identity / reputation / authority). You shall take not in (vain / nothing / worthless / falsehood) His (identity / reputation / authority) who

In other words:

You will be guilty before Yahweh if you reduce your God, Yahweh's authority to nothing. You shall not make His identity false.

or perhaps in other words:

You will be guilty before your God, Yahweh, if you do not submit to His authority, because that will create a false identity of who He is.

Jesus submission

Jesus prayed opened his prayer with...

"Our Father in heaven, hallowed be your name..." -- Matthew 6:9 (ESV)

According to the Interlinear Bible, the Greek for that verse:

Ἁγιασθήτω τὸ ὄνομά σου

which directly translates to

hallowed the name You

Which could be understood as

"Set apart for a special purpose the identity, reputation and authority of You"

or

"I acknowledge that You have an identity and authority which is not like any other"

Opening a prayer this way, means that whatever comes next must be submitted to God's authority to acknowledge his identity. You are not telling God what He should do, you are submitting to God's authority and pray "His will be done" (Matthew 6:10).

"Name" as honor

The word "name" could also mean "honor" as Jesus explains himself:

Then Jesus answered and said to them... “I do not receive honor from men. But I know you, that you do not have the love of God in you. I have come in my Father’s name, and you do not receive me; if another comes in his own name, him you will receive. How can you believe, who receive honor from one another, and do not seek the honor that comes from the only God? -- John 5:41-44 (ESV)