I posted some time back that I had seen a genealogy chart for the Jones family that listed the father of Egbert J. Jones as JOHN. So the middle initial, "J," might be a first name, rather than a last name. I could not remember where I saw that...until now. It's from the Echols family genealogy page:
"Mary Elizabeth (Egbert's wife) is the D/O William ECHOLS IV & Mary Asineth Fennell HOBBS; Egbert is the S/O John Jarrad JONES & Martha WILBURN. The site is: http://echolsfamilygenealogy.blogspot.com/p/env1-marriages-alabama.html

For what it's worth- his middle name might be John or Jarrad. I'm betting on John.