I do not think that anyone becomes an apostle by choice, or seeks out the chance to be in the service of apostles, for that matter.
Barnabas came together with Paul to seek out those who do not know Christ in the wider world. And like Paul he started out with another name, Joseph, and was given a new one "which means, the man of encouragement".
It seems from what we know of him (which is more than most of the original 12) it was an appropriate name. He was a Levite in service to the priests of Aaron, who put himself at the service of the new priests, the Apostles.
Eventually he became the advocate for that other renamed convert, who was such a scourge on the Early Church. Who would ever ask to be in a support position like that? He also seems to have no problem with Paul "outshining" him; he seems content to drift to the background while continuing to spread the Gospel. Yet his gift was that he seemed to sense the potential in others and advocate for them and give them courage to serve.
But he was a man of strong opinions and zeal for the Church and Christ, so it makes sense. It also shows the two sides of that zeal. He breaks with the man he championed over the weakness of another. The faint heart heart of Mark pushed the limits of Paul's human capability to forgive. It is the irony that Barnabas, who forgave and heard the Lord in Paul, should lose to Paul's inability to see and do the same for another. Who would ever wish to be asked to choose one child over another?
Barnabas, from the moment he sold everything and laid it at the feet of the Apostles lived as the man he was, true to his role as Levite and true to his roll as apostle and companion. Eventually this man of encouragement and support even won over Paul's heart. We are fortunate to have Barnabas, who counters both Peter's wishy-washiness and Paul's rigidness.
So it behooves us to emulate him and ask for his constant support and encouragement - perhaps we too can be called "a good man, filled with the Holy Spirit and faith".
The community of believers was of one heart and mind, and no one claimed that any of his possessions was his own, but they had everything in common. With great power the apostles bore witness to the resurrection of the Lord Jesus, and great favor was accorded them all. There was no needy person among them, for those who owned property or houses would sell them, bring the proceeds of the sale, and put them at the feet of the apostles, and they were distributed to each according to need. Thus Joseph, also named by the apostles Barnabas (which is translated “son of encouragement”), a Levite, a Cypriot by birth, sold a piece of property that he owned, then brought the money and put it at the feet of the apostles.
...In those days a great number who believed turned to the Lord. The news about them reached the ears of the Church in Jerusalem, and they sent Barnabas to go to Antioch. When he arrived and saw the grace of God, he rejoiced and encouraged them all to remain faithful to the Lord in firmness of heart, for he was a good man, filled with the Holy Spirit and faith.
-- Acts 4:32-37; 11:21-24