It blew my mind when I first heard the theory that this phrase "under the thigh" was actually a euphemism for "on the loins" or testicles.

In case you think I'm kidding, holding the genitals, or placing the a hand next to the genitals, during the act of solemn oath-taking is attested in several ancient societies. Why?

Well, an oath was considered binding when taken against a "sacred" object, and during those times, the male genitals, which were considered the seat of procreation / source of posterity, were considered sacred and sufficient to make any oath binding.

So here, Abraham may be making his trusted servant swear “on the seed of Abraham” (quite literally) that he would find a wife for Isaac (Gn 24:9).

Another more "Hebraic" interpretation is that Abraham's servant was swearing by touching the sacred of the location of the "covenant of circumcision."

That said, the phrase "on the thigh" is not exclusively euphemistically used.
 
Furthermore, how well these explanations seem to fit Israel’s making Joseph place his hand under Israel's thigh and swear to take his body out of Egypt and bury it where his fathers are buried (Gn 47:29) is up for debate.

For what it's worth though, it's not a coincidence that the English words testimony, testify, testament, and testicle all have the same root.

Here are some links for further reading if you're interested:

https://www.psychologytoday.com/us/blog/games-primates-play/201112/testify-comes-the-latin-word-testicle

https://library.biblicalarchaeology.org/article/put-your-hand-under-my-thigh-the-patriarchal-oath/

https://www.gotquestions.org/hand-under-thigh.html

https://hermeneutics.stackexchange.com/questions/9264/in-genesis-242-why-did-abraham-have-his-servant-make-his-oath-under-his-thigh