Short answer
A Muslim is one who submits to God (﷾). The word Islam means submission, and it shares its root, s-l-m, with salam, which means peace. So a Muslim is one who finds peace by giving himself to the will of the one God (﷾). The Qur'an calls Islam a deen, a word Muslims explain as a complete way of life. In it, giving yourself to God (﷾) shapes belief, worship, character, family and work. The name was not chosen by a teacher, a party or a council. The Qur'an says that God (﷾) Himself named the believers Muslims, "both before and in this" revelation. It uses the same name for Abraham, for those who believed with Moses, and for the disciples of Jesus, because they all submitted to God (﷾). The Prophet Muhammad (ﷺ) told his followers to call themselves by the names Allah gave them: Muslims, believers, servants of Allah (﷾).
Full explanation
What the words mean
Islam is the Arabic word for submission, and a Muslim is one who submits. Both come from the root s-l-m, the three Arabic letters that Islam, Muslim and salam (peace) share. The Qur'an says the deen with God (﷾) is Islam. Deen means religion in the full sense: a complete way of life. The Qur'an promises that whoever submits his whole self to God (﷾) and does good has his reward with his Lord, with nothing to fear.
Gathers the classical Arabic dictionaries. Under the root s-l-m it gives aslama as "he submitted" or "resigned himself", islam as submission, a muslim as one who submits, and salam as peace and safety.
“The Religion before Allah is Islam (submission to His Will): Nor did the People of the Book dissent therefrom except through envy of each other, after knowledge had come to them. But if any deny the Signs of Allah, Allah is swift in calling to account.”
“Nay,-whoever submits His whole self to Allah and is a doer of good,- He will get his reward with his Lord; on such shall be no fear, nor shall they grieve.”
A name given by God (﷾)
The Qur'an says that it is God who named the believers Muslims, before and in the Qur'an. Ibn Kathir, a Qur'an commentator who died in 1373 CE, reports what early scholars said. Ibn Abbas, the Prophet's (ﷺ) cousin, and several others read "He" in this verse as God. They read "before" as the earlier scriptures. A few read "He" as Abraham, whose religion the same verse names. Either way, the name comes from revelation, not from men. The Qur'an also praises the one who calls people to God (﷾), does good and says, in the Arabic, "I am one of the Muslims".
“And strive in His cause as ye ought to strive, (with sincerity and under discipline). He has chosen you, and has imposed no difficulties on you in religion; it is the cult of your father Abraham. It is He Who has named you Muslims, both before and in this (Revelation); that the Messenger may be a witness for you, and ye be witnesses for mankind! So establish regular Prayer, give regular Charity, and hold fast to Allah! He is your Protector - the Best to protect and the Best to help!”
Reports from Ibn Abbas that "He has named you Muslims" refers to Allah, and names Mujahid, Ata, al-Dahhak, al-Suddi, Muqatil ibn Hayyan and Qatada as holding the same view. Mujahid explains that Allah named the believers Muslims before, in the earlier scriptures, and "in this", the Qur'an.
Notes that Abd al-Rahman ibn Zayd ibn Aslam read "He has named you Muslims" as referring to Abraham, since the verse has just named "the religion of your father Abraham", and that Ibn Jarir al-Tabari rejected this, since the verse says they were so named "in this", the Qur'an, which came long after Abraham.
“Who is better in speech than one who calls (men) to Allah, works righteousness, and says, "I am of those who bow in Islam"?”
The name of every prophet's followers
In the Qur'an Abraham and Ishmael pray to be made Muslims, with their children after them. God (﷾) tells Abraham to submit, and he answers that he submits to the Lord of the worlds. Abraham is called neither Jew nor Christian but one who submitted. Moses tells his people to trust God (﷾) if they are Muslims. The magicians who believed with him pray to die as Muslims. The disciples of Jesus bear witness that they are Muslims. So the name the followers of the Prophet (ﷺ) carry is the name the Qur'an gives all who submitted before him.
“"Our Lord! make of us Muslims, bowing to Thy (Will), and of our progeny a people Muslim, bowing to Thy (will); and show us our place for the celebration of (due) rites; and turn unto us (in Mercy); for Thou art the Oft-Returning, Most Merciful.”
“Behold! his Lord said to him: "Bow (thy will to Me):" He said: "I bow (my will) to the Lord and Cherisher of the Universe." And this was the legacy that Abraham left to his sons, and so did Jacob; "Oh my sons! Allah hath chosen the Faith for you; then die not except in the Faith of Islam."”
“Abraham was not a Jew nor yet a Christian; but he was true in Faith, and bowed his will to Allah's (Which is Islam), and he joined not gods with Allah.”
“Moses said: "O my people! If ye do (really) believe in Allah, then in Him put your trust if ye submit (your will to His)."”
“"But thou dost wreak thy vengeance on us simply because we believed in the Signs of our Lord when they reached us! Our Lord! pour out on us patience and constancy, and take our souls unto thee as Muslims (who bow to thy will)!”
“When Jesus found Unbelief on their part He said: "Who will be My helpers to (the work of) Allah?" Said the disciples: "We are Allah's helpers: We believe in Allah, and do thou bear witness that we are Muslims.”
“"And behold! I inspired the disciples to have faith in Me and Mine Messenger: they said, 'We have faith, and do thou bear witness that we bow to Allah as Muslims'".”
How the name Christian began
The name Christian has a different story. Jesus never used it. Acts says only that, years after he had gone, the disciples "were called Christians first in Antioch". Antioch was an ancient city in what is now Turkey.
“And when he had found him, he brought him unto Antioch. And it came to pass, that a whole year they assembled themselves with the church, and taught much people. And the disciples were called Christians first in Antioch.”
The Prophet's (ﷺ) instruction
The Prophet (ﷺ) told the believers to call themselves by the names Allah gave them: Muslims, believers, servants of Allah (﷾). He warned against rallying people under the calls of the age of ignorance, the time before Islam.
Al-Harith al-Ash'ari reports that the Prophet (ﷺ) commanded the believers to hold to the community (al-jama'a), warned that whoever leaves it by a hand-span has thrown off the bond of Islam, and said that whoever raises the call of the age of ignorance belongs to the Fire, even if he prays and fasts. He told them to call themselves by the names Allah gave them: Muslims, believers, servants of Allah.
Perspectives
Mainstream Christian
Christian means a follower of Christ. The name honours the one Christians believe is Lord, and the New Testament uses it with approval.
Peter tells believers not to be ashamed if they suffer as Christians, but to glorify God for it.
Rebuttal
True, the early church honoured the name, but Peter does not say who gave it. Acts says it began in Antioch, years after Jesus had gone. The Qur'an says his own disciples called themselves Muslims, people who submit to God (﷾).
The Islamic perspective
Muslim means one who submits to God (﷾), and God (﷾) Himself gave the name. It belongs to every prophet's true followers, and the Prophet Muhammad (ﷺ) told Muslims to call themselves by the names Allah gave them.
Islam means submission, from the same root as salam, peace.
The Qur'an says the believers were named Muslims both before and in the Qur'an. Ibn Abbas, his student Mujahid, and others read the one who named them as God. Either way, the name comes from revelation.
The Qur'an gives the same name to Abraham, to those who believed with Moses and to the disciples of Jesus.
Submission to the one God (﷾) was the religion of every prophet.
The Prophet (ﷺ) told the believers to call themselves by the names that Allah (﷾) gave them. They are Muslims, believers and servants of Allah (﷾).