Medic canonically has a pet white dove named Archimedes but then, that's a pet, not a daemon. Spider is much more fitting, even despite Archimedes' supposed enthusiasm for chest cavities.

I think its usually jumping spiders that get the "awww, loook at this cute photo of a spider!" treatment. I think it is easier to anthropomorphize them because of their forward facing eyes; they have to keep an eye on where they are jumping!