We'll save you a few minutes of searching right now. You won't find the word neuromancer in any dictionary. Gibson made the word up. Yeah, you can do that. In fact, writers like Shakespeare, Joyce, Tolkien, and The Simpsons guys did it all the time. How else is English going to get great new words to play with? Pretty cool, huh?
To create this word, Gibson took two root words: neuro- and -mancy. Neuro- comes from the Greek neuron meaning nerve and is found at the beginning of words dealing with nerves or the nervous system. Neurological for example.
-Mancy comes at end of words and means "divination" or the act of gaining knowledge through supernatural means. For example, chiromancy means foretelling or divining the future by reading palms while necromancy means gaining supernatural, often unholy, knowledge about the dead. Generally, someone who follows such an art is a magician or wizard within a particular branch of magic. For example, a man who studies necromancy is called a necromancer.
Combining the two root words gives us our title. Basically, it's the title given to a magician who works his magic on the nervous system, gaining knowledge from it, manipulating it, etc.
But why the nervous system anyway? One possible reading is that the nervous system links all five of our senses—touching, tasting, feeling, seeing, and hearing—together. Without our nervous system, we couldn't take in information from the world. We couldn't experience anything. In other words, the nervous system creates our reality.
Gibson's story might be trying to show that by changing how we intake information, we effectively change what can be considered reality, the world around us, and our relationship to it. We see this at play in the novel when Case meets the character Neuromancer, an artificial intelligence who can create a world so real that people can actually live and grow within it.
Of course, on a simpler level, the title is also named for the character Neuromancer. Most titles named for characters tend to be named after the protagonist such as Shakespeare's Macbeth and Mark Twain's The Adventures of Tom Sawyer. So, why not name the book Case? There are a couple possible reasons—besides the fact that Case is a pretty boring title.
First, Neuromancer is a heist story in that it tells the story of a group of people trying to steal something. Typically, although not always, heist movies are named after either the group performing the heist (Ocean's Eleven) or the heist itself (The Italian Job). They can also borrow their title from the object being stolen in the story, such as The Pink Panther (named for the diamond in the film) or National Treasure (Nicolas Cage is after a national treasure after all). While Neuromancer is a character, he's also the object the heist's mission, so he gets top billing dibs.
Second, consider that change in reality we discussed a wee bit ago. Neuromancer needs to be set free if he is going to make that change fully realized. And it's not just Case who is going to experience the change; it possible Neuromancer even changes the entire world—the nature of reality itself. So Neuromancer is a very important character thematically, even if he doesn't get the bulk of page time.
In a way though, it's not very different than Macbeth or The Adventures of Tom Sawyer. In both those works, the title characters are the most important characters thematically. They just happen to be the protagonists as well.