What's the meaning of "神" in "神经"?
There are multiple definitons for 神. One of them can explain why 神经 means nerves/neurons:


  
  心思，心力，注意力：劳～。凝～。～魂颠倒。
  


That is, 神 is used to express "thought", "mental", "attention". Therefore 神经 means "thought/mental/attention pathway", which is a pretty good description of nerves/neurons.

As for why 神 means that as well as things like "god" and "divine", one clue comes from this divining text:


  《皇極經世》天之神棲乎日，人之神棲乎目。


It translates to roughly; "the divinity of heaven resides in the sun, the divinity of man resides in the eye." So you can see how the association between eyes, sight, gaze, attention, thought can be easily made.

People of many cultures have ascribed mystical properties to the eyes and sight. Shakespeare once said, paraphrased, "the eye is the window to the soul." Matthew 6:22 says "the eye is the lamp of the body." (where lamp means light) And what about the all-seeing eye, a representation of God? So the many definitions of 神 provides an interesting view of ancient human philosophy and beliefs.