Tekla’s name day is September 24, the feast of Thekla Equal-to-the-Apostles.
Churches that follow the Old (Julian) Calendar keep the same feast on October 7 of the civil calendar. See everything commemorated on September 24.
The saint behind the name

Thekla Equal-to-the-Apostles
September 24, Holy Woman, Θέκλα
St. Thekla of Iconium was converted as a young woman by the preaching of the Apostle Paul. Renouncing her betrothal to follow Christ, she was condemned to fire and to wild beasts, but tradition relates that God preserved her unharmed; she then labored in preaching and prayer, ending her long life in asceticism near Seleucia.
Also celebrating on September 24
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.