Is there sin upon one who lied to her paternal aunt that her mother is tired, even though the mother is in good health, in order to avoid exposing the girl's leg, which is considered 'awrah in front of non-mahrams, and to avoid inconveniencing the mother?

Lying is forbidden, and obedience to the mother in demanding a lie is not permissible. The questioner must repent from lying. She could have used tawriyah (deliberately ambiguous speech) or ta'reeḍ (indirect allusion) instead of outright lying. Her mother has the right to prevent her aunt from taking the hair from her leg.