Being an (amateur) student of Greek mythology, a hobby that dates back, for me, some 20 years, the only reference I can recall to spiders in Greek mythology is the story of Athena and Arachne.
In the story, Arachne challenged Athena to a weaving competition. Arachne won the challenge, by weaving a tapestry of the Gods depicting them participating in all sorts of wanton acts. This angered Athena (both for the content of the tapestry, and because Arachne's work was superior to her own).
Athena destroyed Arachne's tapestry. Depressed and resentful, Arachne attempted to kill herself (by hanging, if I recall correctly). Athena prevented her death, and turned her into a spider (aka, a "weaver").
While Athena was a goddess of wisdom, a patron of warriors and the arts, like all the Greek gods, had a darker/selfish side. For example, according to the Illiad, it was Athena (by the name Pallas Athene, her first recorded name), who walked between the Trojan and Greek army, inspiring a Trojan (Pandaros, if I recall correctly), to fire an arrow at the Greeks, breaking a fragile peace and re-igniting the Trojan war.
This might be a stretch, and perhaps I'm trying to read too much into this, but this is the only spider-related story I seem to recall from Greek mythology. I'll research it a bit when I get the chance, but perhaps Lord Recluse is related somehow to Athena (representing her warrior side, her prideful side).
