Laura, if you are still reading here after so long of a time in hope of help.

I estimate that your g-grandfather Marvin Abernethy was probably born in the 1880's or '90's, judging from the birth of his son, your grandfather. This would put his father's birth somewhere between 1840's and the 1870's. In my database for North Carolina, I have identified several Henry Abernathy or Abernethys who could be your elusive gg-grandfather. I list them below:

Henry Lynn Abernethy, said to be a son of Miles Abernethy and Nancy Cobb, who was b. abt 1840, but not found on 1850 census records.
Henry L Abernathy, said to be a son of Smith Abernathy & Caroline Jones. This Henry was b. abt 1848 but I have no further information.
Henry Link Abernethy, b. 1 Dec 1862, d 15 May 1925, said to be a son of Albert L Franklin Abernethy & Sophronia Pauline Link. Occupation: medical doctor, buried Oakwood Cemetery, Catawba County, NC. Spouse Fannie Olivia Eckard md 22 Feb 1893.
Henry Walter Abernethy, b. 12 Jul 1866 d. 14 Oct 1931, buried Historic Matthews Cemetery, Mecklenburg Co, NC. Son of John Wesley Abernethy and Martha Ann Rea, ancestor of Bruce, who did not believe it when I found John Wesley's father's will for him because it named the father as Abernathy and not Abernethy.
Henry Clay Abernethy, said to be a son of Joseph Harvey Abernethy & Elizabeth Caroline Sherrill. Spouse Eliza Jane Unknown, no marriage data yet. This Henry was b. 6 Aug 1875 in Mecklenburg County and d. 12 Dec 1948 in Jackson County NC.
Henry Clay Abernathy, said to be a son of Edward Alexander Abernathy & Elizabeth Abernathy from one of the Gaston County Abernathy families. This Henry was b. 9 Nov 1873 d. 18 Dec 1948 in Lincoln County NC, and said to have married 15 Apr 1900 to Violet Eva Goodson.
Henry Miles Abernethy, b. 15 Jul 1889, d. 17 apr 1964. Said to be a son of Marion E. Abernathy & Rebecca Abernethy of a Gaston County Abernethy family. This Henry married twice, first to Bertha Rhyne and then to Beulah Morris. He is buried in the Pisgah Methodist Church Cemetery in Lincoln County NC.
I have no further documentation of candidates born NC between 1840 and 1900.
I hope this may help you.

Regards and apologies for the long delay between your query and my response. I do not normally visit these message bases, but would be grateful for a private email from you in regards to your descent from Marvin. If you have a male cousin or brother who also descends directly from Marvin, I believe I can offer you resolution to your dilemma. Our project has Y-DNA results from two of the above Henrys, and a simple test will show if your male Abernethy relative is a close match or not.

Sincerely, Elizabeth Ferguson
P.S. my email address can be found by clicking on my name on the GenForum message bases for Abernethy and Abernathy, which I check more often than here.