This was also researched by several genealogists who are now deceased, over twenty years ago. They couldn't figure it out either.
My great great grandfather had Don's grandfather living with him when the 1880 census was taken and reported that the young boy was his nephew. Several of his siblings were raised by my family members and a photo of one of his sisters as an adult was even found in the 120 year old Green family bible.
We explored many ways his grandfather could have been related, looking at different families going back to the late 1700s. We couldn't find anything to document a connection.
Now finally we have some proof!
I did a dna test at http://www.23andme.com/ about a year ago and Don did one last month. The results are in and show that we share a common section of Chromosome which means we share a common ancestor. We still don't know who that ancestor was but we are closer. There are three other people in the 23andme database who match us in the same place so we have some real leads which may finally give us the answer.