Hippolyte’s name day is January 30, the feast of Hippolytos.
Churches that follow the Old (Julian) Calendar keep the same feast on February 12 of the civil calendar. See everything commemorated on January 30.
The saint behind the name

Hippolytos
January 30, Martyr, Ιππόλυτος
St. Hippolytos was a martyr of the Church of Rome who gave his life for Christ during the persecutions of the third century. The synaxaria preserve little of his story beyond his faithful confession and his death for the Lord, in company with other martyrs of that city.
Name days: Hippolytos, Hippolyte
Also celebrating on January 30
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.