Eusignios’s name day is August 5, the feast of Eusignios.
Churches that follow the Old (Julian) Calendar keep the same feast on August 18 of the civil calendar. See everything commemorated on August 5.
The saint behind the name

Eusignios
August 5, Martyr, Ευσίγνιος
St. Eusignios was a soldier of Antioch who served long years in the Roman armies, and according to tradition had served under Constantine the Great and beheld the appearing of the sign of the Cross. In extreme old age, in the reign of Julian the Apostate, he was denounced as a Christian; brought before the emperor, he boldly rebuked his apostasy and confessed Christ, for which he was beheaded.
Name days: Eusignios
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.