Can someone convert into Judaism, even if their mother isn’t Jewish?
Theoretically, yes. But most religious authorities will not accept them as Jewish. I know I wouldn’t accept them as Jewish, nor would anyone in my family, nor would any other Jew I know (except maybe Alon).
If they converted halachically (AKA ORTHODOX) then they are 100% Jewish and accepted everywhere. The question itself is confusing. If your mother is Jewish then you do not need to convert. if she is not Jewish then you would need conversion to become Jewish but once that is accomplished (no small task) you are as Jew-y as Jew-y can be.