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.
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.
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.
