You know...I really kind of disagree with some of the advice that you have been given.
It is absolutely true that you cannot "will" your child to a specific party. Its also true that he can't sign documents giving up his rights or agreeing to the child going to your family....or rather, any papers he signed wouldn't be enforceable. However there is also nothing wrong with having him sign the documents anyway since he may choose to honor them.
However, if he is not listed on the birth certificate and paternity has never been legally established then he is not legally the father. If you died social services would not automatically step in and take over and start looking for him. Your child has a family and social services wouldn't necessarily get involved. In fact its fairly rare for them to get involved.
What would happen is that your parents would consult an attorney and file for custody. Since there is no father of record its possible that they would simply be awarded custody. If not, then they might be required to make an attempt to locate him and obtain his consent. If they couldn't, then they would be allowed to serve him via publication.
Its also possible that some time in the future that you will marry, and that your husband may choose to adopt your child. Again, since there is no father of record and the child has clearly been "abandoned" by the bio father again, that would be a fairly simple process.
If you establish paternity and file for child support...neither of those options will be as simple....and the bio-dad might very well want custody in order to terminate the child support obligation.
However child support is your child's right, and completely severing his (or hers, sorry don't remember) connection to the biological paternal family is problematic for many reasons, including practical ones like access to medical history..so those are factors that you also need to consider.
Mull it all over and make your decisions based on what you truly believe is best for your child.