It’s unclear from your query whether you wish to assign custody to another person (e.g., your daughter’s grandmother) permanently through a private adoption or guardianship, informally through a long-term child-rearing arrangement, or conditionally in the event of your death, incapacitation, or other occurrence.
In general, parents (or a sole living birth parent or legal guardian) may transfer custody of their child to a person of their choice. State laws govern the family law issues involved, so the specific legal requirements will depend on which state you live in.
Keep in mind that other people -- particularly an estranged birth parent, the estranged birth parent’s relatives, or a government child services agency -- may dispute your assignment of custody. To ensure that your wishes are fulfilled and everyone’s legal interests -- particularly those of your daughter -- are protected, consult with a local legal clinic or an attorney who specializes in family law. (Ask around for recommendations.) Your desired outcome is too important legally and emotionally to assume the risks involved in handling it on your own.
Disclaimer: This post contains general legal information and should not be construed as legal advice to be applied to any specific factual situation. Each reader should consult a lawyer if you want a qualified professional’s assurance that this information, and your interpretation of it, is appropriate to your particular situation.
I will answer from personal experience with your situation.
My daughter and I agreed that I would assume cusody of her child. The arrangement was to have been temporary, until she was able to change jobs and work hours that would allow her to care for her child in a more appropriate way. Because we lived in different cities, I was not able to provide child care for her.
We live in NC, there was a birth father, but he never saw the child and refused to pay child support, even though there was an agreement as part of their divorce. In NC, we were allowed to go to a lawyers office, have papers drawn up and signed. I then had legal custody of the child.
The arrangement eventually progressed to an adoption. At that time the birth father had to be found and sign over his rights to me. He also was asked to provide a medical history. A home study was done, medical examinations were required, etc. Even though it was an in-family adoption, the paperwork was endless.
If there are other family members with custody rights,i.e. the child's mother, then it is possible that she could challenge your decision even if you have been granted custody. You MUST consult a family law lawyer in any case.