Lydia’s name day is May 20, the feast of Lydia of Philippi.
Churches that follow the Old (Julian) Calendar keep the same feast on June 2 of the civil calendar. See everything commemorated on May 20.
The saint behind the name

Lydia of Philippi
May 20, Holy Woman, Λυδία
St. Lydia of Philippi holds a beloved place in the Church as the first person baptized by the Apostle Paul on European soil. A seller of purple goods from Thyatira, she was a devout worshipper of God, and as she listened to Paul by the riverside at Philippi the Lord opened her heart to receive the Gospel. She was baptized with her whole household and welcomed the Apostles into her home.
Also celebrating on May 20
These names share the feast, many of them other forms of the same name:
What to say on a name day
The Greek wish is Χρόνια Πολλά (Chronia Polla), “many years.” A call or a short message on the day is what people remember. See name day wishes you can send and what a name day is.