What eponymous sign is associated with pain elicited by passive extension of the right thigh while the patient is lying on its side?

The Psoas stretch sign is the correct choice because it specifically refers to the pain that is elicited when the psoas muscle is stretched. This sign is typically assessed in a patient who is lying on their side, and passive extension of the right thigh stretches the psoas muscle, which can trigger pain if there is an underlying issue such as appendicitis or an abscess nearby. This sign is relevant in the context of diagnosing conditions associated with irritation of the psoas muscle or inflammation in the right lower quadrant of the abdomen.

The other signs listed do not apply in this context. Boas sign refers to referred pain to the shoulder due to irritation of the diaphragm, usually from gallbladder disease. Murphy's sign is related to pain upon palpation of the right upper quadrant during inhalation, indicating gallbladder inflammation or cholecystitis. Kehr's sign is the phenomenon of referred pain to the left shoulder due to splenic rupture or other intra-abdominal pathology. Thus, the Psoas stretch sign is distinctly associated with the situation described in the question, aligning perfectly with the symptom elicitation process outlined.