Arachnoid, spider-like, derives from the Greek words for "spider" (arachne) and "like" (eidos). In anatomy, however, the word does not mean "like a spider" but rather "like a spider's web". The arachnoid membrane, the middle of the three membranes that envelope the brain and spinal cord, has a network of exceedingly fine filaments attached to its undersurface (traversing the sub-arachnoid space) that resembles the web of a spider.
The word ultimately goes to back to Arachne, a mythological maiden from Lydia (now western Turkey), who had the audacity to challenge the goddess Athene to a weaving contest. As a result of her brashness, she was killed by Athene (another version of the story has Arachne killing herself) and brought back to life as a spider, forever weaving and forever serving as a warning that mere mortals were not to challenge the immortal gods.