Abimelech (King of Gerar)

Abimelech is the king of the city of Gerar, in the territory of the Philistines, on the border with the land of Canaan. Both Abraham and Sarah, as well as Isaac and Rebecca, spend some time as foreigners in this city.

The name Abimelech means “father is king”. It may well be that this name was passed from father to son, and that all the kings of Gerar had the same name. The king in the time of Abraham would have been the father or the grandfather of the king in the time of Isaac.
Another Philistine king with the name Abimelech is mentioned in Psalm 34:1. The heading for this psalm refers to the story of David acting like a madman in the royal palace. The story itself is set in the palace of King Achish (1 Samuel 21:11-16).
Another person in the Bible with the name Abimelech, is Abimelech the son of Gideon (Judges 8:31).

Abraham and Sarah go to live for a while in Gerar, the city of King Abimelech. Because Abraham is afraid that the king will kill him, so that he can have Sarah for himself, he tells Sarah to say that she is his sister (Genesis 20:1-2).
That does not turn out well. Abimelech has Sarah brought into his palace and into his harem, but God warns him in a dream that Sarah is in fact Abraham’s wife. He has to give her back to Abraham, otherwise God will kill him. The following morning, Sarah is given back to Abraham with many presents. Then Abraham and Sarah leave Gerar. Later Abimelech invites Abraham to form an alliance with him (Genesis 21:22-32).

Years later, Isaac and Rebecca also go to live in Gerar, and what happened before, happens again. Just like his father, Isaac pretends that his wife is his sister, and as before, King Abimelech finds out that this is not true. Although he is very angry, Isaac and Rebecca are allowed to keep on living in Gerar (Genesis 26:6-11).
There, Isaac becomes so rich that the townsfolk become jealous, and Abimelech asks him to go away. Isaac forms a similar alliance with Abimelech as his father had done, and they promise not to harm one another (Genesis 26:26-31).