Why does it say that he fled Sha'ul that day? He fled the previous day!
Radak, Malbim: That day he fled Eretz Yisrael and came to Achish; Sha'ul would not find him there.
Why is he called Achish here, and in Tehilim 34:1, Avimelech?
Rashi (Tehilim 34:1, citing Shochar Tov): his name was Achish. Because he was a Tzadik 1 like Avimelech 2 , he is called Avimelech.
Rashi (Tehilim 34:1): Avimelech is a name for all kings of Pelishtim, just Pharaoh is a name for all kings of Egypt. Radak - the king was called Avimelech also in the days of Avraham and Yitzchak. R. Yeshayah - his name was Achish. Because all the nobles of Pelishtim were under him, he is called Avimelech.
Radak: He had two names. There are many [people] like this.
Yosef Tehilos (Tehilim 34:1, citing the Ari Zal): He changed his Ta'am in front of Avimelech, i.e. he repented to the King of the world.
Shochar Tov (34:1): Galyus' brother told Achish that he wants to avenge the murder of his brother. Achish said no - David killed him in war. Galyus said, "if you can strike me
