Can a paternal grandfather adopt his deceased son's only child in DE

Anything is possible, but you are already his paternal grandfather, so I don't see the point ?

The adoption is so that my granddaugther can receive ssid benefits and benefits because i am a us veteran and her father my son died at age 23 yrs two years ago and we were told that she would not get any benefits at all from her dad, my son so that is why i want to adopt her but we want to do it privately as possible between the child's mother and mysekf that is why i needed to know thank you

Its not likely that the courts would allow an adoption. But they would allow custody. What I don't understand is what the tie in is to receiving benefits? Can you explain that some more.