[The Squatter of Amon Rûdh]

I think he meant the story of Arachne the weaver who was turned into a spider, by Athena i believe.

I see now. The girl who, forced into a weaving competition with Athena by her father's bragging, won, then hanged herself to escape the Olympian's wrath. Athena, took pity on her by turning her into a spider (one advantage of being a god is that you don't have to make sense).
I'm still not entirely clear about the Ungoliant link. Are we considering the possibility that Ungoliant's form was some sort of punishment? I've always assumed that she just liked looking like a repulsively bloated spider, or that it matched her personality so well that it was the easiest form for her to assume.