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

Samuel the Prophet
August 20, Prophet, Σαμουήλ
The Holy Prophet Samuel was the last of the judges of Israel, born in answer to the prayers of his mother Hannah, who dedicated him to God from childhood. Serving in the sanctuary at Shiloh, he heard the call of the Lord as a boy, and grew to be a faithful prophet, judging Israel in righteousness.
Also celebrating on August 20
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.