The most famous name in the story is not in the product's name at all. "Corn Flakes" just describes what is in the box. The fight was over who got to put "Kellogg" in front of it.
The cereal began as an accident with wheat. In 1894 the brothers John Harvey Kellogg and Will Keith Kellogg were working at the Battle Creek sanitarium. History Today says they were trying to make an easily digested bread. Wikipedia's account is that John left a batch of wheat-berry dough behind. He later ran it through rollers and got thin flakes that could be baked.
They called those wheat flakes Granose. Who did what is disputed, and Wikipedia says the family's accounts tend to play down each other's roles. Etymonline credits the discovery to W. K. Kellogg, but the patent, issued in 1896, named John alone. It covered many kinds of grain, not just wheat.
Corn came next. History Today dates flakes of toasted maize to 1898, and a longer-lasting version to 1902. Etymonline records "corn flakes" for the breakfast cereal from 1902, though the phrase had been used in the 1890s for an ingredient in brewing.
The word "corn" here is American. In Old English it meant grain or a single seed, any cereal at all. Since about 1600, Etymonline says, American English has used it for maize, first as "Indian corn." In England "corn" usually means wheat, and in Scotland and Ireland, oats.
"Flake" is older and plainer. It meant a thin flat piece of snow from the early 14th century. Its origin is uncertain, perhaps Old English or Old Norse flak, "flat piece."
The family name arrived with the business. Will, the sanitarium's business manager, added sugar to make the flakes appeal to a mass audience, which caused a rift with his brother. He founded the Battle Creek Toasted Corn Flake Company on February 19, 1906. It later became Kellogg's.
Both brothers used the Kellogg name, and they fell out over it. Will won a 1911 lawsuit for exclusive use of the name in the United States. A second legal fight, from 1916 to 1921, extended that to other countries.