Did you catch the grammar of that sentence? They were called Christians. They didn't call themselves Christians. In fact, up until this time, they had called themselves by very ordinary names: believers, brothers/sisters, disciples, followers of the Way. But in Antioch, they were called Christians. It was not a name they gave themselves, but one given to them by outsiders based on what they observed in their lives.
So what did they observe about this church? The rest of the chapter gives us some clues.
Persistent in spite of persecution (Acts 11:19): The church in Antioch began with believers fleeing persecution in Jerusalem that had begun with the death of Stephen, but they did not give up their fledgling faith. They kept believing in spite of the risk.
Radical in their evangelism (Acts 11:20): Antioch was a large and prosperous city. It had a Jewish population, but it was predominantly Gentile. Some of the believers fleeing persecution began sharing the good news of Jesus with the non-Jews in Antioch. This may not seem like a big deal today, but at that time the idea that you could have faith in Jesus and NOT be Jewish was still quite radical.
Passionate in their faith (Acts 11:21): Several times, we are told that in response to the preaching of the gospel, a great number of people believed and turned to the Lord. For Gentiles to do so meant turning away from a background of idolatry and very likely a more licentious lifestyle. But they had discovered the truth and turned to the Lord.
Generous in their giving (Acts 11:27-30): As soon as the Gentile Antioch church knew of another church's coming trouble, they wanted to help - even though it was a church with a different cultural background than their own (Jewish believers in Jerusalem).
Sensitive to the Spirit (Acts 13:1-3): It was the church at Antioch that commissioned the first missionaries, Saul (later known as Paul) and Barnabas. Their calling as missionaries, however, was not independent of the church. Rather, it was made clear in a context of worship, fasting, and prayer, all of which created an environment and an opportunity where they could be sensitive to how God was leading them, both corporately and personally.
Humble in heart (Acts 15:1-2): After some time, teachers came to Antioch who tried to convince the church that they must not only believe in the Gospel, but adhere to certain Jewish practices, to be saved. There was quite a bit of disagreement before the church settled on sending Paul and Barnabas to Jerusalem to talk it over with the apostles and elders. They didn't just bully their way into a decision, but submitted themselves to those in authority over them and trusted their situation to God.
Grounded in God's Word (Acts 15:35): Over and over, we see that different people stayed in Antioch to teach and preach the word of the Lord, the gospel. These were people hungry for the truth and willing to devote time to learning what the Word of the Lord was.
Do you see what the people of Antioch noticed about these people whom they derisively called the Christianoi? They saw people who were crazy about Jesus, who were committed students of His Word, who cared about right theology, but were also humble and generous, who didn't let persecution sway them from their message that Jesus is the only way back to the Father. And they saw people who loved each other, as family.
But if we want to be called Christians by others - well that seems to be about being crazy about Jesus, trusting Him with your whole eternity, letting Him mold your character, and being so convinced of who He is that nothing stops you from sharing that good news with the people around you. If we can do that well, as the church of Antioch did, then we will really be able to impact the world for good, for eternity.