How long after puberty onset should my daughter start the birth control pill?

Can start now: There isn't generally a need to delay in initiating the pill after puberty. I assume she is using it to regulate her cycles or to address cramping or menstrual symptoms.

Not in early puberty: The hallmark of entering puberty in girls is breast development. If you start ocp on a girl just after pubertal onset she will fuse her growth plates fairly quickly and lose final height potential. We don't start girls on birth control (except under specific circumstances, like primary amenorrhea) before menarche. Breast development may also be negatively impacted by introducing ocp too soon.