Where is the father? You cannot just give up custody without consulting with the father of the child. Your child does not get to decide where she wants to live. You certainly can allow the grandmother to have guardianship, but you will need the father's permission or it will have to go to court. Have you browsed through the information in LawInfo's Free Legal Resource Center to learn more about your issue yet? See:
http://www.lawinfo.com/consumer.html and
http://resources.lawinfo.com/en/index.html. You can certainly try to speak to a lawyer to determine what legal options may be available. In the meantime, you may be able to learn more on your own. Search the "Free Legal Resources" tab, or browse the Consumer Resources. Good luck.