What does it mean when it says he put his hand under Abraham's thigh? Is this a custom?
The commentators have different opinions about the meaning over here: According to Rashi, based on the Midrash Rabbah, it does not mean literally the thigh; it means the Milah (organ of circumcision). The reason is because one who takes an oath must hold in his hand a sacred object, such as a scroll of the Torah or phylacteries. And the circumcision was his (Abraham’s) first commandment and came to him through suffering. And it was beloved to him. And (therefore) he chose it (as the object upon which to take the oath). This is also the opinion of Tosefot in the Talmud Shevuot 38b. The Keli Yakar also learns that he made him swear with the Milah. However he adds an annotation explaining the deep meaning of this action. That due to the circumcision a man becomes restricted from forbidden relations. That is why he said don’t take a wife for my son from the daughters of Cana’ani, because lust is their inheritance from Cham the father of Cana’an. Some say he mated Noach [Sanhedrin 70a]. Therefore it is not fitting for the circumcised to associate with the lustful. Due to the circumcision God blessed Abraham, and due to Cham mating Noach Cana’an was cursed, and the blessed do not adhere to the accursed. According to the Ibn Ezra it means the thigh literally. Thus that was the custom in those days, for a man to put his hand under the one who rules him. Meaning if you are under my authority please put your hand under my thigh. And the master sits with his thigh on the hand. Meaning it is as if he is saying, here my hand is under your authority to fulfill your wishes. This custom still exists in India (in the time of the Ibn Ezra). This is also the opinion of the Rashbam and the Chizkuni. See also the Netziv in Ha’amek Davar and the Maharal in Gur Aryeh.