Solomon Albertson was born 1754 in Pitt county, TN; he was the son of John Albertson and Elizabeth Early. He had a brother Early born 1756, and a sister Ann. This has been recently discovered by Diana Garcia in the Herd family Bible. More details are available on the Alberson/Albertson Family website; I will send all three of you an invitation.
The name Alberson was simply a variation on the name Albertson. Solomon's last name was written as Albertson in Pitt county deeds, but later as Alberson in some of his census records.
Solomon had sons Early and Moses. Their census records are:
Solomon: 1790, 1800 and 1810 in Pitt, NC; 1820 in Wayne, TN
Early: 1800 in Pitt as Yerley, 1820 in Wayne, TN; 1840 in Lafayette co, MO; 1850 and 1860 in Miller co, MO
Moses: 1830 and 1840 in Wayne, TN; 1850 and 1860 in Perry, but probably did not move. Rather, when Perry was created from Wayne, Moses was in the "Perry" part
The composition of Moses' household shows a male 0-10 in 1830 and 10-15 in 1840. This was probably John,as the ages are consistent with his age of 30 in the 1860 Census.
Early (1780) moved to Lafayette; one of his sons, Solomon born 1819, moved with his family to Oregon by 1850. This branch of the family is well documented. I would guess that the fact that John named his son Henderson Lafayette might be related to the fact that his uncle Early had been in Lafayette county.
Anyway, I believe that Michael Lafate's information is excellent, and quite conclusive: Henderson Lafate, also known as Henderson Lafayette, was the son of the John, born 1830, who was still in Perry county, Tennessee for the 1860 census.
Doris: In that census, John's birthplace is clearly identified as Tennessee, not Virginia (I just checked the Images Online, and am quite certain of this).
Michael: You note that your grandfather was named Franklin. He is probably the James W F who was born in September 1884. He is listed as a son of Henderson in the 1900 Census for Bollinger county, Missouri.