The man known as Arphax Dawson, who was killed in the 1862 event known as "the Great Hanging" of Gainesville, Texas, had the given name of ARPHAXAD. His name was absolutely NOT ARPHAXTON, as is mistakenly claimed on literally thousands of web-pages and compiled genealogies which have been perpetuated in error over the years.
Below are all the various spellings I have found of Arphax's name and the documents they came from. I have never seen a well-written record showing Arphax's name to be Arphaxton. The 1860 Census record for his household records Arphax as "A.R." and one of his sons as "Anfaxton" (closer to Anfaxton than Arfaxton). This is the only record I've found that could come close to leading someone to say that Arphax's name was Arphaxton. There may be others, but I have not found them or seen them referenced in any sources I can find.
As to Arphax's son in the 1860 census, who is reasonably claimed to be "Arphax, Jr.", one of his descendants has told me that he went by A.R. Dawson, Jr. and that his nickname was Dan. This descendant referred to him as ARPHAXED (Deana Dawson Link, great-granddaughter of Earl Monroe Dawson, who was a son of Arphax, Jr.)
There are numerous cases to indicate that Arphax was named after the biblical Arphaxad (see Wikipedia article link below). There is no doubt in my mind that Arphax's name was Arphaxad or Arphaxed, leaning heavily toward Arphaxad. I can see there could be a reasonable argument for Arphaxed, since there are instances where that spelling or a close variation was used (Araphaxed, Arofaseed), but the most frequent spelling I can find is Arphaxad, and most all of the badly written documents could be construed to have meant Arphaxad and not in any case, Arphaxton.
I am open to other interpretations of the information conveyed below and very much would like to learn of other sources where Arphax's name is displayed. But as for me, the following representations of Arphax's name, in combination with the likely origin of his name (the Bible), lead me to conclude his name is Arphaxad Dawson.
And so here is the documentary evidence used for determining the given name of Arphax Dawson. I'll present each spelling found in chronological order:
####AROFASEED (1830) Source: 1830 US Census, Population Schedule, Jackson County, Alabama, Page 106; NARA Series: M19; Roll Number: 1; Family History Film: 0002328. 
####ARAPHAXED (1834) Source: United States Land Patent, Bureau of Land Management Automation Project, White County, Illinois, Accession # IL0070__.328, Patentee for Parcel at 3rd Prime Meridian, Twp 5-S, Range 10-East, Section 34,W½NE¼, issued August 7, 1834. see: [http://www.glorecords.blm.gov/details/patent/default.aspx?accession=IL0070\_\_.328&docClass=STA&sid=20xi2lqh.pdw#patentDetailsTabIndex=1](http://www.glorecords.blm.gov/details/patent/default.aspx?accession=IL0070\_\_.328&docClass=STA&sid=20xi2lqh.pdw#patentDetailsTabIndex=1)
####ARPHA??, ARPH?A?D, AR?HA?A?AD (1837) Source: - Jackson County, Alabama - Deed Book A, p 147-148 (Deed from Heirs of William Dawson to Arphax Dawson). Three mentions in same document (1st two times are split between 2 lines)
1. 
2.
3.
####ANA'SON / ARA'SON (1850) Source: 1850 US Census, Population Schedule, Franklin County, Illinois, Stamped Page 44, dwelling 609, family 626. 
####ARPHAXAD (1851) Source: United States Land Patent, Bureau of Land Management Automation Project, Franklin County, Illinois, Accession # MW-1045-177, Patentee for Parcel at 3rd Prime Meridian, Twp 7-S, Range 1-East, Section 21, SE quarter, issued March 1, 1851. see: http://www.glorecords.blm.gov/details/patent/default.aspx?accession=1045-177&docClass=MW&sid=n2dssurh.ktr#patentDetailsTabIndex=1 
####ARPHAXAD (1853) Source: United States Land Patent, Bureau of Land Management Automation Project, Franklin County, Illinois, Accession #IL1870__.405, Patentee for Parcel at 3rd Prime Meridian, Twp 7-S, Range 1-East, Section 21, E½NE¼, issued September 10, 1853. see: [http://www.glorecords.blm.gov/details/patent/default.aspx?accession=IL1870\_\_.405&docClass=STA&sid=n2dssurh.ktr#patentDetailsTabIndex=1](http://www.glorecords.blm.gov/details/patent/default.aspx?accession=IL1870__.405&docClass=STA&sid=n2dssurh.ktr#patentDetailsTabIndex=1) 
####ARPHAXAD (1853) Source: National Archives, Land Entry File. Three different documents associated with the 1853 Land Patent (just above) in Franklin County, Illinois. 1.  2.  3. 
####ARPHAXAD (1859) Source: Grayson County, Texas - Deed Book "L", page 152 (typed transcription); Jacob M. Huffaker sale to Arphaxad Dawson. 
####ANFAXTON (1860) The name of Arphax's son, presumed to be a Jr., was written as "Anfaxton" in the 1860 Census. Arphax, Sr. was referred to as "A. R". in this household.
Source: 1860 US Census, Population Schedule, Grayson County, Texas, Sherman P.O., page 180b, sheet 94, dwelling 431, family 432
####ARPHAXED / APHAXED (1867) Source: Cooke County, Texas - Deed Book 7, pages 226-22, dated July 4, 1867. Sale of land to Henry C. C. Crow, by heirs of Arphax Dawson. This parcel straddles the Cooke and Grayson County line and is just north and west of Tioga, Texas. It was part of (and perhaps all of) the original John Fizer (Heirs) Survey. 
Others:
ARFOXED
In "Early Land Grants of White County (1814-1976?) and Probate Index 1818-1976" by Harriet B. Vaught, page 81, "Arfoxed Dawson" is listed as administrator for the estate of "Doreas Phillips" (Dorcas?) [Year not noted].
ARFAX
In "Cooke Co., Texas: Where the South and the West Meet", p. 12, it mentions that Arfax Dawson and Jim Ward fled their homes and sped to where Mill Creek empties into the Red River, finding the water too high . . . later that morning rode into Gainesville and turned themselves in.
####GENERAL INFORMATION ABOUT THE PERSON FOR WHOM I BELIEVE OUR ARPHAX WAS NAMED:
Arpachshad or Arphaxad or Arphacsad . . . was one of the five sons of Shem, the son of Noah (Genesis 10:22, 24; 11:10-13; 1 Chron. 1:17-18). His brothers were Elam, Asshur, Lud and Aram; he is an ancestor ofAbraham. He is said by Gen. 11:10 to have been born two years after the Flood, when Shem was 100. http://en.wikipedia.org/wiki/Arpachshad
####INFORMATION ABOUT THE GENERAL IDEA OF ANYONE IN THE WORLD BEING NAMED ARPHAXAD OR ARPHAXTON . . .
Finally, I can find no person EVER, anywhere in the world, to have possessed the given name of Arphaxton, or any use of that word besides on the internet by genealogists claiming it for a given-name for our Arphax Dawson.
I have found over a half-dozen or so people named Arphaxad both currently and in the past:
Arphaxad Ashley, died 1920, Syracuse, NY
Arphaxad Carroll, Jr., University of Idaho track athlete (originally from Las Vegas NV)
Arphaxad Githui (from Linkedin.com-a citizen of Kenya)
Arphaxad Thompson (b. Woburn, Mass. 7 March 1763)
Arphaxad Kemmerlin, a high school football player in Summerville, SC
--and used as a middle name
James Arphaxad Key, born in 1788 at Giles County, VA
There are a similar number of instances of people with the name Arphaxed:
Arphaxed Loomis, a lawyer born in Winchester, Connecticut, 9 April, 1798
Arphaxed Jacob Dawson, 1879-1962 of Clark County, Missouri
--and used as a middle name
John Arphaxed Aminga
Loring Arphaxed Williams of Glenwood, Iowa
But NOT SO for Arphaxton. Googling the world "Arphaxton" brings only repeated instances by prolific genealogists perpetuating the non-person of Arphaxton Dawson.