Sorry to hear about the Lynch Syndrome.
I don't know about ovarian cancer (although if your doctor says so, he's probably right), but I know bc pills can help prevent uterine/endometrial cancer.
A lot of endometrial cancer starts as endometrial hyperplasia, which is when the uterine lining grows too thick and the cells become abnormal. This condition is usually caused by a hormone imbalance- either too much estrogen or not enough progesterone (leaving the estrogen unopposed).
Either birth control pills or progesterone supplements alone are often given to women with this condition, to regulate their cycles and cause the uterine lining to shed regularly. This prevents the development of uterine cancer.
Good luck with everything, and kudos to you for being proactive and vigilant.