“Now Abraham was old, advanced in age; and the LORD had blessed Abraham in every way. Abraham said to his servant, the oldest of his household, who had charge of all that he owned, “Please place your hand under my thigh…”” Gen. 24:1-2
Genesis chapters 23 and 24 are largely comprised of Abraham getting specific affairs in order before his death, namely, securing a family burial site, and finding a suitable bride for his son Isaac.
As Abraham is known as a man who does many things himself (such as making preparations for the journey to Moriah on his own in Gen. 22:3, even though servants could have done this for him; see also Gen. 18), it may seem odd that he is entrusting a servant with the momentous task of finding a bride for Isaac.
We note of course Abraham is advanced in age (v. 1), and quite possibly on his deathbed. Although we read about Abraham taking another wife in Genesis 25, we must remember that there is often overlap between chapters and that the narrative does not always flow chronologically (consider the accounts of creation in Genesis 1 and 2 for example.) Additionally, there is a good chance that Abraham has already passed away by the time the servant returns, as we see in Gen. 24:65 that the servant now refers to Isaac as his master. (Oxford)
Although it is awkward to speak of, the phrase “place your hand under my thigh” is most likely referring to the male organ or testicles. Today we take oaths on a sacred object, such as a bible. Jews may swear on a t’fillin or a Torah scroll (Plaut). According to Rashi, the fact that God asked Abraham to circumcise himself, or perhaps the importance of the seed of Abraham and the promises it held for the future, made the male organ the most significant object to place hands on.