"a spider's web," early 14c., coppewebbe; the first element is Old English -coppe, in atorcoppe "spider," literally "poison-head" (see attercop). Spelling with -b- is from 16c., perhaps from cob. Cob as a stand-alone for "a spider" was an old word nearly dead even in dialects when J.R.R. Tolkien used it in "The Hobbit" (1937).
Figurative use for "something flimsy and easily broken through" is by 1570s. Plutarch attributes to Anacharsis, the 6c. B.C.E. Scythian-born philosopher in Athens, the statement, variously given, that laws were like cobwebs that entangled the little flies but wasps and hornets never failed to break through them. An old Norfolk term for a misty morning was cobweb-morning (1670s).
"spider," Old English atorcoppe "spider," literally "poison-head," from ator "poison, venom" (Middle English atter), from Proto-Germanic *aitra- "poisonous ulcer" (source also of Old Norse eitr, Old High German eitar "poison;" German eiter "pus," Old High German eiz "abscess, boil;" Old English atorcræft "art of poisoning") + copp "top, summit, round head," probably also "spider" (compare cobweb and Dutch spinne-cop "spider").
Amptes & attircoppes & suche oþer þat ben euere bisy ben maide to schewe man ensaumple of stodye & labour. [Elucidarium of Honorius of Autun (Wycliffite version) c. 1400]
It lingered in Northern England dialect in the sense "peevish, ill-natured person" (c. 1500).
"net, snare," 1520s, from French toile "hunting net, cloth, web" (compare toile d'araignée "cobweb"), from Old French toile "cloth" (11c.), from Latin tela "web, net, warp of a fabric," from PIE root *teks- "to weave," also "to fabricate." Now used largely in plural (as in caught in the toils of the law).