Eliud’s name day is July 20, the feast of Elijah the Prophet.
Churches that follow the Old (Julian) Calendar keep the same feast on August 2 of the civil calendar. See everything commemorated on July 20.
The saint behind the name

Elijah the Prophet
July 20, Great Feast, Ηλίας ο Προφήτης
The Holy Prophet Elijah the Tishbite lived in the ninth century before Christ, in the days of King Ahab of Israel. Zealous for the Lord against the worship of Baal, he shut the heavens by his prayer, was fed by ravens at the brook Cherith, raised the son of the widow of Zarephath, and called down fire from heaven on Mount Carmel. He was taken up alive toward heaven in a chariot of fire, leaving his mantle to his disciple Elisha.
Name days: Ilias, Elias, Elia, Elijah, Ilya, Eliud, Iliana, Ilianna
Also celebrating on July 20
These names share the feast, many of them other forms of the same name:
Ilias, Elias, Elia, Elijah, Ilya, Iliana, Ilianna
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.