"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).