Is your daughter willing to let you adopt or have full custody of the child? That would be the first obstacle. My parents went through three changes of custudy when my sister would come back and say she wanted to be mom again... finally, the court made it a finalized adoption done deal... since the judge knew the circumstances... but that was the judges decision. In order to have the birth mother out of the picture for awhile, you might have to move or have strict rules? There may also be a lot of hard feelings from the child toward the mother...especially if more kids come along later and get preferential treatment.
You'd also need to consider if the biological father is in the picture at all... or if he is known with name on birth cirtificate etc...
If you adopt the child versus full permanent custody, it might make a difference as far as government assistance of even school enrollment information... you'd want to talk with an attorney to hear the different aspects of those options.
Keep encouraging the granddaughter not to have bad feelings against the mother... try to be possitive and hopefully when the granddaughter is old enough to understand she can learn that the biological mom wasn't emotionally ready to be a parent at the time the child was born... not as a condemnation of her mother, but as an explanation so that she understands that she was not responsible or in any way doesnt feel that she was not good enough for the mother to want her... those things can have tremendous impacts on adolescents...and hopefully a life lesson about the importance of waiting to have children will be learned by the youngest generation...
There is also an issue of other family members spreading untrue information through gossip that can cause a lot of friction in the family as well as from other members of the community who may only hear one side of things. Some of those things are impossible to figure out who said what to whom... or how the stories get started... just be aware that those things do happen... but it really doesn't matter because you know the truth... let others say or think what they choose... they probably will no matter what you might say...if you ever even have an opportunity to say anything....
Sorry if I've made any wrong assumptions or written too much... I've just seen this type of situation play out for many many years now...