Tanya’s name day is January 12, the feast of Tatiana.
Churches that follow the Old (Julian) Calendar keep the same feast on January 25 of the civil calendar. See everything commemorated on January 12.
The saint behind the name

Tatiana
January 12, Martyr, Τατιανή
St. Tatiana was a deaconess of the Church of Rome who suffered martyrdom in the third century. Raised in a devout home, she served the Church in prayer and in care for the sick and the poor. Arrested in a time of persecution, she confessed Christ with courage and endured many tortures without denying her Lord.
Also celebrating on January 12
These names share the feast, many of them other forms of the same name:
Tatiane, Tatiani, Tania, Tatiana
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.