The evidence that Sampson D. Fisher was son of Elijah Fisher is circumstantial:

The listing for Elijah Fisher in the 1810 Christian Co KY census showed one male <10. Both Jacob and Hiram were born in 1815 or later in MS.  In addition the 1820 census records Elijah Fisher family with 5 males (meaning two extra sons) and the 1830 Hinds Co census records Elijah Fisher with 7 males (meaning four unknown sons).

Sampson D. Fisher could well have been named after Sampson Dees, who was father of Elizabeth Dees Fisher.

Sampson’s son William Hudson Fisher named children Mary Catherine, Anna Elizabeth, and David--all names of other children of Elijah Fisher. He also named a son “Johnson Deeds Fisher”, which could well have been after his great grandfather Sampson Dees.

Sampson D. Fisher went to Texas in 1839 and purchased land in Shelby Co TX in 1839 and in Rusk Co TX in 1846:

Unconditional Cert. Shelby Co TX, 5 dec. 1839
Conditional Cert. Shelby Co TX, 7 Dec. 1846
Unconditional Cert. Rusk Co TX, 1846

He is listed in a Panola Co TX census of 18461555 and in the 1850 and 1860 federal censuses of Panola Co TX. In 1850 he is listed as having been born in MS and in 1860 he gave his state of birth as KY.

More on Sampson D. Fisher:

February term Probate Court 1843
San Augustine County, Texas
Estate of William Fisher
The petition of Sampson D. Fisher having been filed praying for Letters of Administration upon the estate of Wm. Fisher diceased (sic) and no objections having been made.
It is therefore ordered by the Court that the said estate of Wm Fisher dicd.(sic) by his giving bond & security in the sum of two thousand dollars &...........
There being no further business Court adjourned
March Term Probate Court 1843
San Augustine County
Estate of Wm. Fisher
It is ordered by the Court that the petition of Sampson D. Fisher for letters of administration on the estate of William Fisher decd be dismissed from said Court. In consequence of the said S. D. Fisher non compliance with the orders of the Court & that execution issure against the said Sampson D. Fisher for costs ........
 
1850 Panola County Texas
115. 115 House number
Samson D. Fisher     age 40  born in Mississippi
Catherine                31              Mississippi
Wm.                       8               Texas
 
1850 Panola county, Texas
102.102 House number
Wm Fisher 25 Alabama
Sarah A. 24 MISS
William   3 MISS
 
Probate records of Panola County, Texas
#116 - William Fisher, deceased - Administration - January 1852 Term of
Court - Other people mentioned in the papers: S.D. Fisher

From the research of Notha Stevens:

Sampson D Fisher is not proven to be the son of Elijah and Elizabeth Dees Fisher. At this time there is only circumstantial evidence:

1. His name is one item of circumstantial evidence... Elizabeth Dees who married Elijah Fisher, no doubt is the daughter of Sampson Dees who died in Warren County, Mississippi in 1817.

2. Sampson D b. ca 1810 , is only found on 2 census's , one he shows he was born in Miss., one he shows b. in Kentucky. Elijah Fisher was in Ky in 1810. There are very few Fisher males in Miss. in 1810. I beleive he was born in Christian County, Kentucky.

3. His son William Hudson named his son Johnson Deeds Fisher (is this an indication that Sampson's initial "D" was Dees/Deeds?)

4. We have evidence of a relationship of Elijah Fisher and Penny Fisher Sessums, because of their ages it has been determined they were probably siblings. Jacob Sessums is living with son Redding Sessums in Shelby County, Texas in 1850, two sons and one daughter are living in Panola County, Texas in 1850, not far from Sampson D. Fisher. Sampson was one of the administrators for the estate of Penny's son, Soloman Sessums in Panola County, Texas

5. Redding Sessums and Sampson Fisher both received land in Shelby County, Texas in December 1839

6. Sampson Fisher married in Madison County, Mississippi in 1838, checking the map and the location of Elijah's land you will note how close Elijah lived to the border of Madison County

7. Penny Fisher Sessums descendants have family tradition information about her being an Native American Indian. Sampson D. Fisher descendants have this same family tradition. It doesn't appear as any of the other Mississippi families have any information about Indian blood

8. Penny Fisher Sessums descendants and the descendants of Sampson D. Fisher went to Indian Territory Oklahoma to register for Indian benefits

9. Item to be concerned with placing Sampson as the son of Elijah Fisher (but no doubt there seems to be a relationship) is contact with a William Fisher. His probate is in San Augustine, Texas in 1843, with Sampson D. Fisher applying as adm. No further record has been located on this probate. A William Fisher signed with Almon Robbins on his marriage to Catherine Fisher, proven daughter of Elijah. It appears this William Fisher may have a son named Willaim b. ca 1825 in Alabama.  1850 is in Panola County, Texas and Sampson D. Fisher was involved in his estate in Panola County. This is where I am with the research at this time.