The name was originally going to be Fighting Fantasy, but due to concerns over trademark conflicts with the roleplaying gamebook series of the same name, they needed to settle for something else. As the English word "Final" was well-known in Japan, Sakaguchi settled on that.
Now, why were they going to be fighting tictacs? That I don't don't know.
When the first one was in development, the lead designer was going leave the games industry and go back to university if it didn’t do well. In addition, SquareSoft was on the brink of bankruptcy, meaning this game could very well be the final game either of them were going to release.
This is an urban legend. They were originally going to title the game "Fighting Fantasy," but they couldn't due to the Steve Jackson fantasy game book series having that name already. They liked the "Fi Fa" sound, so they came up with "Final" for the alliteration and because it sounded cool.