The 120 gathered believers proposed 2 people: Joseph called Barsabbas (also called Justus) and Matthias. They were presumably both suitable. They prayed that God's choice should be shown and then they cast lots,
"...and the lot fell on Matthias".And that's really all we know about him. He's not mentioned by name again in the New Testament. Perhaps Theophilus, for whom Luke wrote the Acts of the Apostles, already knew a lot about Matthias, so didn't need any extra details? Various traditions recount that Matthias was an evangelist and it is rumoured he was eventually martyred.
who in the place of the traitor Judas
chose your faithful servant Matthias
to be of the number of the Twelve:
preserve your church from false apostles
and, by the ministry of faithful pastors and teachers,
keep us steadfast in your truth;
through Jesus Christ your Son our Lord,
who is alive and reigns with you,
in the unity of the Holy Spirit,
one God, now and for ever. Amen.
Collect for Matthias, Apostle, Common Worship