It is not a Checkers variant, nor did it originate in China, how could this name possibly make sense?
There seems to be surprisingly little information about the origin of the name "Chinese Checkers". The game itself was first released in 1892 in Germany as "Stern-Halma" (literally "Star-Halma"), indicating that it was a variation of Halma that was played on a star-shaped board. But there doesn't seem to be any solid information about why the name "Chinese Checkers" became associated with the game.
The only thing that I could find was a history of the game on PlayChineseCheckers.com, which despite sounding authoritative is written by an anonymous author who doesn't back its claims (not to mention has a website that looks like a GeoCities page from the 90s) and so the following passage should be taken worth a grain of salt:
The first Chinese Checkers game to be published in the United States was 'Hop Ching Checkers' in 1928 by J. Pressman & Co. This was exact the same game as the 1892 Star-Halma. The brothers Bill and Jack Pressman made up the name 'Chinese Checkers' during or shortly after 1928. The game was given a Chinese name and theme in keeping with the current interest in all things oriental (among them the discovery of King Tut's tomb in 1922 and the 'mah jongg' game that was introduced in 1923)
I'd like to stress again that despite this being a really solid-sounding explanation, the author is anonymous and there are no sources to back up this claim, so it's not clear whether or not this is true.
After a great deal of searching, I did not manage to find a more authoritative source on the subject, so the true origins of why the game is called "Chinese Checkers" may remain a mystery.