Abraham is old, and the question of Isaac's wife must be settled. In Targum Pseudo-Jonathan on Genesis 24:2, the Aramaic makes explicit what the Hebrew only hints at: Abraham tells Eliezer to put now thy hand upon the section of my circumcision.
The Hebrew says, put your hand under my thigh — a euphemism that ancient readers understood and modern readers often miss. The Targum of Pseudo-Jonathan names the organ directly: the sign of the covenant, the berit milah.
This is the most sacred oath-object Abraham possesses. Not the altar. Not the well at Beersheba. His own flesh, marked when he was ninety-nine years old (Genesis 17:24), carrying the physical sign of the covenant with heaven. To swear by it was to invoke the covenant itself as witness.
Eliezer is the most trusted servant in the household, described in the Torah as the zekan beito — elder of the house — who ruled over all of Abraham's property. He is going on the most important diplomatic mission of Abraham's life: finding a wife for the son on whom every promise depends.
The Maggidim read this verse as showing that the deepest oaths are sworn on what cost you the most. The takeaway: when you need someone to understand that the stakes are ultimate, bring them into contact with the covenant you carry in your own body.