Tina’s name day is July 24, the feast of Christina the Great Martyr.
Churches that follow the Old (Julian) Calendar keep the same feast on August 6 of the civil calendar. See everything commemorated on July 24.
The saint behind the name

Christina the Great Martyr
July 24, Martyr, Χριστίνα Μεγαλομάρτυς
St. Christina the Great Martyr suffered for Christ at Tyre in the third century. Born to a prominent pagan family and kept apart by her father, she came to knowledge of the one true God, and destroyed the idols of gold and silver set before her, giving the fragments to the poor. For this she endured imprisonment and cruel tortures at the hands of successive judges, remaining unshaken until she received the crown of martyrdom.
Name days: Christina, Christiana, Kristina, Chris, Tina
Also celebrating on July 24
These names share the feast, many of them other forms of the same name:
Christina, Christiana, Kristina, Chris
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.