The word “mnemonic” has nothing to do with anything in the movie. Nothing. It’s just a weird word that makes the title sound fancy.
As an adjective, “mnemonic” means “aiding or designed to aid the memory” or “relating to the power of memory”. As a noun, it means a special word or poem that helps you recall a set of connected ideas—like “FANBOYS”, which reminds you of seven coordinating conjunctions in English: for, and, nor, but, or, yet, so.
But the premise of Johnny Mnemonic is that the protagonist is carrying data in his head like a drug mule and doesn’t even know what it is! Moreover, he doesn’t remember much about himself; he dumped his memories to make more space to carry data. From the title, I would have guessed he had a special memory skill, but no. He just has a cybernetic upgrade that turned his brain into a (rather faulty) hard drive. He’s nothing special. Might as well be named John Doe.
William Gibson wrote the screenplay as well as the short story of the same name, so the title was his choice, nothing to do with Hollywood. Maybe he picked it because the character wishes he could remember his childhood?
See below for more on what the movie is like and why I didn’t particularly like it.