Circumstances change everything, so much of what happens will depend on your story.
My kids are adopted. The adoptive parents tell them. In our case, we told them they were adopted way before they were old enough to even know what it meant. That way it was just something they always knew and thereafter we answer questions as they are asked.
The items would be left with whomever has the baby. An exception to that might be if there was an elderly parent of the deceased who just was too old or frail to care for a baby, but took care of the child for a few days/weeks until it was adopted. She may keep the items as mementos of her own child, to be given to the baby upon her death. That is considering you had an "open" adoption.
If both her parents died, then more than likely next of kin would show up for the baby. You might have a battle between mom's side of the family and dad's side of the family. My best guess in this instance would be family members, if not them then the State would take custody, place the baby in foster care, as Monkey said, then adopt after time passed and red tape was worked through.
The adoptive parents names are on the birth certificate after the adoption is final, with no indication that the child has been adopted. Someone told me that many years ago, birth certificates of children who had been adopted had a corner snipped. I have no idea if that is true. The original certificate, however, would probably remain in the hands of whomever has the box, unless it was given to the courts for proof of baby's birth information, where it would be filed in the case (which would be sealed).
