Sarah was afraid of the negative influence which Yishmael might have on Yitzchak. She saw Yishamel “playing.” She said that Hagar his mother and Yishmael must be sent away because Yishmael can not inherit from Avraham with her son Yitzchak.
Their are several opinions about what Yishmael was playing with that made Sarah want to get rid of him. Some say that he was involved in Idolatry, others that he was involved in licentious behavior and others say that he was involved in murder.
The Tana Rabbi Shimon bar Yochai says that it is impossible to entertain the thought that such behavior occurred in the same home as Avraham Avinu. Instead, the games which were being played were ordinary sibling rivalry ones. Yishmael felt that he was the first born so he would receive an inheritance double the size of Yitzchak. When Yitzchak was born people were happy that Avraham now had a son who can inherit from Avraham. Yishmael said that Avraham already has a first bor son. When Sarah found out she decided the son of her Maid Servant had gone too far.
The book, Torah Temimah discusses how half brothers, like Yishmael and Yitzchak are often bitter rivals when it comes to inheritance from their mutual father.
We should take into consideration the Statement of Our Rabbis of Blessed Memory who said that Maaseh Avot Siman Lebanim – The action of the forefathers is a sign of what will happen with the children. Yishmael and Yitzchak made the claim over the same inheritance. G-d had decided that Yitzchak will inherit the land of Canaan and that Yishmael will also father many nations.
Before Yishmael was born Sarah had oppressed Hagar. She did this because it was improper that a maid should look down upon her mistress. In her despair she escaped from her oppression. G-d sent an angel to console her. He told Hagar that she would have a son whom she would call Yishmael. He was Called Yishmael because G-d had heard her Suffering.
The Midrash, Perkey Derabi Eliezer has a very interesting additional explanation on the name Yishmael. It says that G-d will hear the suffering of the nation [of Israel] when oppressed by the children of Yishmael. Although throughout history the descendants of Yishmael have oppressed the Jewish people they have never before tried to conquered Israel from the Jewish people. One version of this Midrash says that this oppression will be “in the end of days.”
Towards the end of our Perasha of Chaye Sarah we see Yitchak and Yishmael together with seemingly no animosity. They got together in the city of Chevron to bury and mourn the death their father Avraham Avinu.