Name Day is in calendars around the world the feast day of the saint after whom a person is named at baptism. In early centuries, Christians began honoring the first saints - also known as martyrs - who were slain for their faith. Each year, on the anniversary of the martyr's death, Christians celebrated the day as a feast day, the martyr's birthday, into heaven. It was more of an occasion of rejoining, rather than a day of mourning.
See a Greek Nameday calendar here.