Maura’s name day is May 3, the feast of Timothy and Maura.
Churches that follow the Old (Julian) Calendar keep the same feast on May 16 of the civil calendar. See everything commemorated on May 3.
The saint behind the name

Timothy and Maura
May 3, Martyr, Τιμόθεος και Μαύρα
The Holy Martyrs Timothy and Maura were a young Christian couple of Egypt who suffered during the persecution of Diocletian at the end of the third century. Timothy was a reader of the Church, entrusted with the sacred books; only days after their marriage the two were arrested when he refused to surrender them. Together they endured cruel tortures and were crucified, encouraging one another in the faith to the end.
Also celebrating on May 3
These names share the feast, many of them other forms of the same name:
Timotheos, Tim, Timothy, Mavra, Mavre
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.