Arachnophobia is the fear of spiders, and other arachnids such as scorpions. It is a word developed from the two Greek words arachne, "spider" and phobos, "fear."
This fear is something that all children of Athena suffer from. Though other demigods may share the fear, it is not as deep as it is for children of Athena. In ancient times there was a girl named Arachne who was a beautiful weaver of tapestries. Arachne soon started to boast to others that she was better at weaving than Athena, the goddess of weaving, herself. Athena was angry that a mortal would dare to compare her work to that of the gods so she gave her a chance to redeem herself.
Athena went down to earth disguised as an old lady and warned Arachne not to offend the gods. Arachne didn't care and told her that Athena herself could have a weaving contest with her. Angrily Athena threw off her disguise and accepted the challenge. Arachne created a tapestry detailing foolishness of the gods and their cruelties towards the mortals. Athena created a tapestry praising the gods for their achievements. When the weaving was finished Arachne continued to brag saying hers was better. Athena, in a fit of rage turned Arachne into a spider and ripped her tapestry before anybody saw it.
From that point on, the children of Athena fear spiders because they worried that each spider was Arachne coming to take revenge on them. The modern terms Arachnid and Arachnophobia are both derived from this ancient Greek myth.
When Hephaestus' metal spiders attacked Percy Jackson and Annabeth Chase, Annabeth panicked. She later revealed to Percy that she had an immense fear of spiders due to her mother's history with Arachne. Percy had noted that Annabeth behaved differently when she encountered the spiders, because of her fear. She started to panic, but managed to regain control of herself.
Annabeth was scared of the metal spider that led them to Daedalus. However, she still followed it, albeit hesitantly. When it landed near her feet, she flinched and yelped.
Annabeth Chase encountered spiders multiple times (and Arachne herself) during The Mark of Athena. She is described to have felt very intense terror. Arachnophobia is also included in several of her flashbacks contained in the book.