Chayei Sara | Place your Hand under my Thigh - Solving a Bereishit Riddle

When Avraham prepares to send his slave to find a wife for Yitzhak, he makes a curious demand of him: "Put your hand under my thigh" either as a prefatory act to an oath or as part of the oath itself. This request/demand appears in only one other place in Tanakh - when Yaakov has Yosef put his hand under his father's thigh before committing to bury his dying father in Canaan. What is the meaning of this odd practice? Is it rooted in near eastern oath customs? Connected to the Halakhic requirement of holding a sanctified item while swearing? Is it a sign of servitude? We survey the approaches of the Rishonim - including some surprising observations - and then offer a new wrinkle to solve this age-old conundrum.