Elisha’s name day is June 14, the feast of Eliseos the Prophet.
Churches that follow the Old (Julian) Calendar keep the same feast on June 27 of the civil calendar. See everything commemorated on June 14.
The saint behind the name

Eliseos the Prophet
June 14, Prophet, Ελισσαίος
The Holy Prophet Elisha lived in the ninth century before Christ. Called from the plough by the Prophet Elijah, he left all to follow him, and when Elijah was taken up in the fiery chariot Elisha received his mantle and a double portion of his spirit. He prophesied in Israel for many years, working great wonders: healing the waters, multiplying the widow's oil, cleansing Naaman the leper, and raising the dead.
Also celebrating on June 14
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.