When Jesus' movement got underway following the remarkable event of Easter, the movement called itself "The Way." No doubt it comes from Isaiah 40:3 'prepare the way of the Lord in the wilderness, prepare the way of the Lord' and that's what they were doing. John the Baptist had appealed to that very verse in his ministry. In fact, we even find it in one of the Dead Sea scrolls, in one of the very important books from Qumran. They understood themselves as "the way" as foretold in Isaiah 40. We see that reference several times in the book of Acts - so if you wanted to refer to Christians during the early Christian movement you referred to "those who belong to The Way."
But, at Antioch people started calling them Christians. What's interesting is that Antioch was north of Israel - modern day Lebanon - and it was outsiders who started calling them Christianoi (Kris-T-ah-noy). That doesn't really separate Christians from Jews or Judaism anymore than being called a Pharisee meant you are no longer a Jew or a Sadducee or an Essene. So Christianity being given a name meant "within the community of Israel among the Jewish people, the followers of Jesus - who was regarded as the Christ, the Messiah - are now called Messianists or in Greek Christianoi.
So that's where the name came from - so even then - well into the book of Acts, decades into the history of the church - the name Christian really meant a Jewish group that is fascinated with and centered on Jesus of Nazareth who is regarded as the Messiah or in Greek, the Christ, the anointed one.
So the name Christian - even in the first century - did not mean non-Jewish. To be a Christian in the first century nobody would think, "Oh! He's a Christian - that's strange - I thought he was a Jew - I guess he's not a Jew anymore." Nobody thought that way. To be a Christian was to be a Jew.
I don't know about you but I love understanding the origin of things, especially our Christian roots.