"Jackson" was Greg's mother's maiden name. Greg lived most of his life with his father's legal name, Daniel, and used it often on his early internet websites, most notably his 2005 GregoryJDaniel.com website, "J" for his middle name James.
During most of his YouTube career, Greg had gone by "Gregory Jackson" online. During this time, when Greg was asked about his real name, he would usually say something like "[My real name is] Gregory Jackson (according to my Mom)". (Source)
Legal Name Changes
Greg first attempted to change his legal last name on 2010 to Sicesca. Sicesca was a religion he made up and was following for a few years In his divorce paperwork he requested the change, but did not seem to go through with it. In 2012, when Greg married Kai, they both changed their last names to Avaroe.
In 2019, Greg and Kai changed their last name to Jackson and dumped their middle names. Greg changed his first name to James and Kai changed his legal first name to Kai. Shortly after changing his name to James Jackson, Greg uploaded a video announcing his name change. He says he will no longer go by Greg and wants to be called "JJ". ($ource)
Other Past Use
Greg would often use the name "Jackson" as a source of encouragement and strength during his military days, which can be seen in his 2005 online journal:
"I realize many cry and complain for being so emotionally abused in boot camp, yet I am convinced no such tears will reach my eyes like the others often have ~ I am full of Jackson strength, and devotion to my cause, you can all have faith in that."
"Deep down, I am not just a Jackson, I am the son of a great woman, she is the true Jackson I see, that and her father ~ the two people I respect most, one because I know her well, I've seen so many honorable things... and two because the one I respect looks up to another, and no doubt, if the one I look up to looks up to another, than I can easily call such a person also my superior, my idol." (Source)