|"Elizarus and Rebekah at the Well," Johann Carl Loth, 1670's|
You have to love an account of courting between a man and a woman which begins with a man telling another man, "Put your hand under my thigh..."
But, Biblical Times. So. In the words of 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.
(Rabbi Moshe Leib Halberstadt at Yeshiva.org)
Now that this is out of the way...
Read Genesis 24, and notice what a folk take it is...
Now Abraham was old, well advanced in years; and the Lord had blessed Abraham in all things. Abraham said to his servant, the oldest of his house, who had charge of all that he had, “Put your hand under my thigh and I will make you swear by the Lord, the God of heaven and earth, that you will not get a wife for my son from the daughters of the Canaanites, among whom I live, but will go to my country and to my kindred and get a wife for my son Isaac.”
The portions of scripture that are most easily linked to the oral sources tend to be repetitive. This story contains instructions from Abraham to his servant (unnamed, the "oldest of his house," like Abraham... so, Abraham's true surrogate) on procuring a wife for Isaac. No Canaanite women, but a woman from his country and his kindred (just like his sister-wife, Sarah).
The servant travels, says a prayer for a sign and for success, encounters Rebekah (at a spring! Perhaps a post on wells and springs would be good. Have you noticed they pop up in stories concerning women?), engages her, says a prayer of thanksgiving, and goes in to meet her father Bethuel and her brother Laban.
The servant then tells, almost verbatim, the same story we have been told, only to the family of Rebekah.
But let's talk about Rebekah.