Basiliskos’s name day is May 22, the feast of Basiliskos.
Churches that follow the Old (Julian) Calendar keep the same feast on June 4 of the civil calendar. See everything commemorated on May 22.
The saint behind the name

Basiliskos
May 22, Martyr, Βασιλίσκος
The Holy Martyr Basiliskos suffered for Christ in Pontus at the beginning of the fourth century, during the persecutions of the pagan emperors. Tradition remembers him as a kinsman of the Great Martyr Theodore the Recruit and a soldier like him. Refusing to sacrifice to the idols, he endured imprisonment and cruel treatment, and was beheaded at Comana, faithful to the end.
Name days: Basiliskos, Vasiliskos
Also celebrating on May 22
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.