This question can be answered by taking a look back evolutionary and see how other organisms have utilized neurology. The prime example, because it was one of the first real models of the neurological system, is the squid. It has neurons with singularly long axons to send electrical pulses long distance. This method of electric pulse transfer has been evolutionary updated, essentially, but we can get an idea of how these cells came to be. If we look back at squid predecessors, like the jellyfish, we see that it also developed electrical impulse cells to aide in defense and movement. Back further, we see the sponge has no neurons, but has something similar. This gives us potential insight into the development of the cell. It has protein clusters that act similar to post-synaptic signaling. So it is likely the electrical impulses were developed based on the function of these protein clusters, and bam! Electrical pulses became a must faster way to communicate to different regions of a system than simply trying to protein signal across cells.
Of course, looking at this question evolutionary is tricky since the question of divergent evolution comes into play, and we can't know if the sponge protein clusters offer a potential answer to neuronal development. But, is interesting evidence and makes an interesting debate!
Source= Sakarya O, Armstrong KA, Adamska M; et al. (2007). Vosshall, Leslie, ed. "A post-synaptic scaffold at the origin of the animal kingdom". PLoS ONE