Who first coined the term "Christian"?

It looks like the Church in Antioch may have been officially 'founded' in 35 AD according to the Orthodox church and it may have consisted of Jews who had believed the Gospel. Then around 42 AD non-Jewish believers began to be accepted. Sometime later Barnabas sends to Tarsus for Paul to come and teach. Paul leaves Antioch on his first missionary journey around 45 AD. Luke records in Acts that it was during Paul's one year stay in Antioch that the disciples were first called Christians.

So Christians got their name around 44 AD.

We have Ignatius born in Syria (likely Antioch) in 35 AD and, traditions say, converted at a young age. Ignatius would have been around 9 years old when the phrase 'Christian' was coined. Not impossible.

There doesn't seem to be any reason to believe the folks in Antioch were calling themselves by that name. Most often we see them referring to one other as 'saints' in the New Testament and in early writings. If Ignatius were already converted it is likely not him. There is an idea that the term Christian was originally a term of derision, which would make Ignatius unlikely to have begun it's use unless it was before his conversion: also not impossible. However, it is also possible that the society in Antioch (referred to as 'all the world in one city') merely needed a label for this new thing that was happening: See this interesting article. This would make Ignatius an unlikely candidate as a 9 year old socialite.

So...Did Ignatius coin the phrase which may or may not have been derogatory? It is unlikely but not impossible.

It was St Evodius of Antioch, the successor of Peter and predecessor of St Ignatius who coined the term Χριστιανός or Christianos, modern term, Christians.

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Here are some commentaries on the term Christian.

"The disciples were called CHRISTIANS first in Antioch." This name originated not within, but without, the Church; not with their Jewish enemies, by whom they were styled "Nazarenes" ( Act 24:5 ), but with the heathen in Antioch, and (as the form of the word shows) with the Romans, not the Greeks there [OLSHAUSEN]. It was not at first used in a good sense (as Act 26:28 1Pe 4:16 show), though hardly framed out of contempt (as DE WETTE, BAUMGARTEN, &c.); but as it was a noble testimony to the light in which the Church regarded Christ--honoring Him as their only Lord and Saviour, dwelling continually on His name, and glorying in it--so it was felt to be too apposite and beautiful to be allowed to die. Jamieson, Fausset, Brown

[6.] Thus the scripture was fulfilled, for so it was written (Isa. 62:2) concerning the gospel-church, Thou shalt be called by a new name, which the mouth of the Lord shall name. Matthew Henry

v. First called Christians can also have the idea that they were called Christians before they were called anything else. They first identity was now to be called Christians. Today, Christians must be willing to take at least the idea of the title “Jesus People,” and must also be worthy of the name. Instead of claiming any other title – Roman Catholic, Protestant, charismatic, whatever – we should be first called Christians.

vi. Eusebius, the famous early church historian, described a believer named Sanctus from Lyons, France, who was tortured for Jesus. As they tortured him cruelly, they hoped to get him to say something evil or blasphemous. They asked his name, and he only replied, “I am a Christian.” “What nation do you belong to?” He answered, “I am a Christian.” “What city do you live in?” “I am a Christian.” His questioners began to get angry: “Are you a slave or a free man?” “I am a Christian” was his only reply. No matter what they asked about him, he only answered, “I am a Christian.” This made his torturers all the more determined to break him, but they could not, and he died with the words “I am a Christian” on his lips. (Eusebius, Church History) David Guzik

Who first coined the term “Christian”?

The short answer is we just do not know. It is doubtful that it was St. Ignatius of Antioch!

It is true that the term Christian was first employed at Antioch; there exist no historical evidence as to who coined this term.

The first recorded use of the term (or its cognates in other languages) is in the New Testament, in Acts 11 after Barnabas brought Saul (Paul) to Antioch where they taught the disciples for about a year, the text says: "[...] the disciples were called Christians first in Antioch." (Acts 11:26). The second mention of the term follows in Acts 26, where Herod Agrippa II replied to Paul the Apostle, "Then Agrippa said unto Paul, Almost thou persuadest me to be a Christian." (Acts 26:28). The third and final New Testament reference to the term is in 1 Peter 4, which exhorts believers: "Yet if [any man suffer] as a Christian, let him not be ashamed; but let him glorify God on this behalf." (1 Peter 4:16).

Kenneth Samuel Wuest holds that all three original New Testament verses' usages reflect a derisive element in the term Christian to refer to followers of Christ who did not acknowledge the emperor of Rome. The city of Antioch, where someone gave them the name Christians, had a reputation for coming up with such nicknames. However Peter's apparent endors*m*nt of the term led to its being preferred over "Nazarenes" and the term Christianoi from 1 Peter becomes the standard term in the Early Church Fathers from Ignatius and Polycarp onwards.

The earliest occurrences of the term in non-Christian literature include Josephus, referring to "the tribe of Christians, so named from him;" Pliny the Younger in correspondence with Trajan; and Tacitus, writing near the end of the 1st century. In the Annals he relates that "by vulgar appellation [they were] commonly called Christians" and identifies Christians as Nero's scapegoats for the Great Fire of Rome. - Etymology of the term Christian (Wikipedia)

Although we do not know who actually coined the term Christian for those who follow The Way; it quite possible that it was coined by the pagans of Antioch.

St. Ignatius of Antioch is however accredited with coining the term Catholic.

The earliest recorded evidence of the use of the term "Catholic Church" is the Letter to the Smyrnaeans that Ignatius of Antioch wrote in about 107 to Christians in Smyrna. Exhorting Christians to remain closely united with their bishop, he wrote: "Wherever the bishop shall appear, there let the multitude [of the people] also be; even as, wherever Jesus Christ is, there is the Catholic Church."

Of the meaning for Ignatius of this phrase J.H. Srawley wrote:

This is the earliest occurrence in Christian literature of the phrase 'the Catholic Church' (ἡ καθολικὴ ἐκκλησία). The original sense of the word is 'universal'. Thus Justin Martyr (Dial. 82) speaks of the 'universal or general resurrection', using the words ἡ καθολικὴ ἀνάστασις. Similarly here the Church universal is contrasted with the particular Church of Smyrna. Ignatius means by the Catholic Church 'the aggregate of all the Christian congregations' (Swete, Apostles Creed, p. 76). So too the letter of the Church of Smyrna is addressed to all the congregations of the Holy Catholic Church in every place. And this primitive sense of 'universal' the word has never lost, although in the latter part of the second century it began to receive the secondary sense of 'orthodox' as opposed to 'heretical'. Thus it is used in an early Canon of Scripture, the Muratorian fragment (circa 170 A.D.), which refers to certain heretical writings as 'not received in the Catholic Church'. So too Cyril of Jerusalem, in the fourth century, says that the Church is called Catholic not only 'because it is spread throughout the world', but also 'because it teaches completely and without defect all the doctrines which ought to come to the knowledge of men'. This secondary sense arose out of the original meaning because Catholics claimed to teach the whole truth, and to represent the whole Church, while heresy arose out of the exaggeration of some one truth and was essentially partial and local.

By Catholic Church Ignatius designated the universal church. Ignatius considered that certain heretics of his time, who disavowed that Jesus was a material being who actually suffered and died, saying instead that "he only seemed to suffer" (Smyrnaeans, 2), were not really Christians. Historical use of the term Catholic (Wikipedia)

Side note:

According to an ancient tradition, St. Ignatius of Antioch was the child whom Christ took and presented to the apostles as the example of the one who is greater in the kingdom of heaven. From that day the child, who was most beloved by the Savior and favoured with the divine embrace, was also marked as the one upon whom lions would feast in the Roman Colosseum.

At that hour the disciples came to Jesus, saying: Who thinkest thou is the greater in the kingdom of heaven? And Jesus calling unto him a little child, set him in the midst of them, And said: Amen I say to you, unless you be converted, and become as little children, you shall not enter into the kingdom of heaven. Whosoever therefore shall humble himself as this little child, he is the greater in the kingdom of heaven. (Matthew 18:1-4)