Jude’s name day is June 19, the feast of Apostle Jude, Brother of the Lord.
Churches that follow the Old (Julian) Calendar keep the same feast on July 2 of the civil calendar. See everything commemorated on June 19.
The saint behind the name

Apostle Jude, Brother of the Lord
June 19, Apostle, Ιούδας Θαδδαίος
The Holy Apostle Jude, called Thaddeus, is numbered among the Twelve and is known as the Brother of the Lord, being of the household of the righteous Joseph. He wrote the short catholic epistle that bears his name, urging the faithful to contend earnestly for the faith once delivered to the saints. After Pentecost he preached the Gospel in the lands of the East, and tradition holds that he ended his labors as a martyr.
Also celebrating on June 19
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.