First question: Do you have a clue about his parent's names? I have found several possibilities...not documented but they might provide you with a direction to head in your digging.
I've found several trees that list his father as William Charles Cochran b. 1770-75ish. married to Elizabeth Brock b. 1772ish. These two are out of South Carolina according to the record.
Another possible is a William Winn Cockerham (this is one variation for Cochran) married to a Nancy Estes. No location on these two.
You might also try the spelling Cockeran since I found Elisha with that spelling same birth year at least once.
Again no documentation that I could find on this in the short time I looked at it but may help direction wise....for William Charles Cochran parents listed are Alexander Cochran b. 1738ish married to Elizabeth Camp b. 1745ish. Alexander is listed as coming from Paisley, Renfrewshire, Scotland. (Note found an alternative location on another tree: Stoneykirk, Wigtown, Scotland and this one had a baptism record attached)Elizabeth from North Carolina.
Alexander's parents are listed as Alexander Cochran b. 1701 and Margaret Buchanan b. 1709. Both from Paisley Abbey, Renfrewshire, Scotland. Found an alternative (see above) with the mother listed as Janet McGaw
One more possibility and then I am off to bed...lol
Same progression...William Charles Cochran married to Elizabeth Brock both out of South Carolina...William's father being Alexander Cochran b.1742 but this time from Chester, Pennsylvania died in South Carolina and Alexander's father being James Cochran b. 1698 from Mayboy, Londonderry, Ulster, Ireland died 22 May 1768 in West Fallowfield, Chester, Pennsylvania, United States
I also randomly found one tree with his father listed as Charles Weatherford out of North Carolina b.1720.
Sorry I couldn't find anything concrete or documented but hope that if you didn't have these potentials before that they might be useful in heading you in the right direction.