What was the relationship between Lydia and Paul?
By the river at Philippi, Lydia listened to Paul, and the Lord opened her heart. After she was baptized, she welcomed him into her home. Acts 16:13-15
What the Bible says
Acts 16:14
And a certain woman named Lydia, a seller of purple, of the city of Thyatira, one that worshipped God, did hear: whose heart the Lord opened to attend to those things which were said by Paul.
Acts 16:15
And when she was baptized, and her household, she besought us, saying: If you have judged me to be faithful to the Lord, come into my house, and abide there. And she constrained us.
Douay-Rheims Bible (Challoner revision), public domain.
What the Church teaches
The Catechism recalls the whole households that were baptized, like Lydia’s.
Catechism 1252 · Catechism of the Catholic Church