There was a particular set of criteria that qualified Matthias to be the next apostle.
In the Acts of the Apostles, St. Peter discusses who should replace Judas. He then begins to go through a set of criteria before praying to God for guidance, ultimately choosing Matthias.
In a homily of St. John Chrysostom that is featured in the Church’s Office of Readings, we learn there were very specific requirements that the replacement needed to meet before they were even considered.
Lived in our company
So, he goes on, we must choose from those men who lived in our company. Notice how insistent he is that they should be eyewitnesses. Even though the Spirit would come to ratify the choice, Peter regards this prior qualification as most important. Those who lived in our company, he continued, all through the time when the Lord Jesus came and went among us. He refers to those who had dwelt with Jesus, not just those who had been his disciples. For of course from the very beginning many had followed him.
It was important for Peter that the replacement be someone who was there with them and who similarly walked with Jesus and learned directly from him.
Witness of the resurrection of Jesus
Up to the day, Peter added, on which he was taken up from us – one of these must be made a witness along with us of his resurrection. He did not say “a witness of the rest of his actions” but only a witness of the resurrection. That witness would be more believable who could declare that he who ate and drank and was crucified also rose from the dead. He needed to be a witness not of the times before or after that event, and not of the signs and wonders, but only of the resurrection itself. For the rest happened by general admission, openly; but the resurrection took place secretly, and was known to these men only.
Being a witness of Jesus’ resurrection was an important qualification, especially for a young Church that needed to convince others that Jesus was raised from the dead.
They needed strong leaders, those that had a firm belief in Jesus and could personally attest to his resurrection from experience.
In the end, they left the choice of Matthias to the providence of God, not relying on their own power, and he was chosen by God to replace Judas.