It is considered a disgrace not to know who your father is and having "illegitimate" stamped on your birth certificate let the whole world know the circumstances of your birth. Anney knows that this word indicates shame and that Bone will be branded with it all her life if Anney doesn't get it removed.
As long as the birth certificate has that demeaning word on it, Bone will never be able to escape her past and start over. An unbranded birth certificate symbolizes Bone's rebirth.

It's also ironic that Anney is so intent on having Bone's birth certificate changed when she allows Glen to abuse her most of her life. At the end, Anney deserts Bone and then gives her the "clean" birth certificate. In her mind, Anney is giving Bone a gift of taking away the stigma of her birth. Anney has allowed Glen to take away Bone's innocence and childhood, and Bone will be branded with that abuse for the rest of her life.