Acts 16:1 Then came he to Derbe and Lystra: and, behold, a certain disciple was there, named Timotheus, the son of a certain woman, which was a Jewess, and believed; but his father was a Greek: 2 Which was well reported of by the brethren that were at Lystra and Iconium. 3 Him would Paul have to go forth with him; and took and circumcised him because of the Jews which were in those quarters: for they knew all that his father was a Greek.
Again, we are not told why Paul chose Timothy.
We might speculate the following:
1. Paul knew Timothy’s faithful character.
2 Timothy 1:5 When I call to remembrance the unfeigned faith that is in thee, which dwelt first in thy grandmother Lois, and thy mother Eunice; and I am persuaded that in thee also.
2. Timothy was a Jew on his mother’s side. He knew the scriptures, and by circumcision, would be able to accompany Paul when preaching in Synagogues.
5 Timothy 3:15 And that from a child thou hast known the holy scriptures, which are able to make thee wise unto salvation through faith which is in Christ Jesus.
3. Paul imparted to him a Holy Spirit gift by the laying on of his hands
2 Timothy 1:6 Wherefore I put thee in remembrance that thou stir up the gift of God, which is in thee by the putting on of my hands.
4. Paul could trust him
1 Corinthians 4:17 For this cause have I sent unto you Timotheus, who is my beloved son, and faithful in the Lord, who shall bring you into remembrance of my ways which be in Christ, as I teach every where in every church.
I hope you find this helpful.