Most Star Wars fans know that the name "Starkiller" dates back to some of the original scripts for what eventually became A New Hope. At one point, one of the main characters was a young woman named Starkiller. Later, she was transformed into an older male general. Eventually, as the story was reworked, Mark Hamill played Luke Starkiller, and in early shoots, he introduced himself to Leia as such. It was reshot, possibly because the name had too negative of a connotation, and "Skywalker" was cemented into canon, while "Starkiller" faded into legendary status.
In 2008, LucasArts ruined my life by making me buy my first video console in almost 10 years with the release of The Force Unleashed. The main character, Galen Marek, was, at one point, an agent of Darth Vader operating under the code name "Starkiller." Eventually, he turned to the light, and died in a failed attempt to defeat Palpatine. His efforts allowed some senators to escape from danger and establish The Alliance to Restore The Republic, and adopt the Marek family crest (starbird) as the symbol. "Starkiller" had made its way into canon in a big way.
It will always be "Bigger Death Star" to us.
The Great Disney Canon Reboot of 2014 once again changed the status of the name. It relegated The Force Unleashed to non-canon Legends, and along with it, the exploits, and name, of "Starkiller."
In 2015, Disney likely ended any chance of the character of Starkiller returning to the canon, when the superweapon in The Force Awakens was named Starkiller Base. Apparently, this was considered a better name than "Bigger Death Star," but it wasn't our call. Unless the name of the base was inspired by a character named Starkiller that is as yet uncovered in future canon, it would seem that the character most of us know as Starkiller was gone forever.
However, there is this story, in which it would seem Dave "I Am Canon" Filoni almost wrote Starkiller into the animated series Rebels. According to the article from comicbook.com, Sam Witwer (Darth Maul in both animated series, as well as Starkiller in the video games) said via his Twitch stream:
Dave [Filoni] did share with me, by the way, he considered making Starkiller an Inquisitor in Star Wars Rebels. It didn’t quite fit the story that they were telling, but he did think about it because he thought that would be interesting.
Now, I don't know what the hell a Twitch stream is, but it sure sounds important enough that he should see a specialist immediately. Nevertheless, it seems Filoni, who wields canonical power, could, at any time, re-canonize Starkiller. Perhaps he was just thinking of making Galen Marek an Inquisitor and dropping the code name, since, as mentioned, all your "Starkiller" are belong to base. Or something like that. Maybe Filoni had worked out some way for this inquisitor to be the inspiration for Snoke's superweapon, or perhaps the name would just be common in the Galaxy, like Antilles. Or Raymond Luxury Yacht.
So there you have it. While canned for the moment, there is at least a chance that we might see Galen "Starkiller" Marek re-canonized. It's a long, winding history for the name, and there may yet be more to come.