My mother told me certain oral contraceptives did not wok for her, will this be the same for me?

No! With no missed pills, "perfect use", the failure rate of birth control pills is 0.3%, that is >99% effective. In typical use, because taking a pill every d without ever missing is hard, about 9/100 couples will get pregnant/year. Not missing pills is the best way for the pills to work for you.
No. The definition of "work" with birth control pills is vague, it usually means not getting pregnant. For them to not work can be because they weren't taken every day or other meds were taken that decrease their effectiveness, such as antibiotics. This is not genetic. If "not working" was having side effects, like mood changes, to some types of hormones, that may or may not happen to family members.