Neurons evolved from amoeba-like cells. Amoeba are protists that are known to produce pseudopodia, which are protrusions from the cell body that allow the cell a degree of motility. The theory goes that these protrusions, which involve the internal restructuring of microtubules, at some point became permanent, resembling dendrites and axons that are today characteristic of neurons. For more information, see Antonio Damasio’s Self Comes to Mind (Vintage, 2010).
See the pseudopodia of the amoeba on the left, and the pyramidal neurons on the right.