You could always resurrect some pet names from the past: Charlotte (more appropriate for a spider than a cat anyway, but maybe too obvious), or Snowy (just because it's so totally inappropriate for a spider). 

Or:  Muffet, Animal (from the Muppet Show), Santa's Little Helper (just to drive the zealots crazy), or Spot.  Somebody I know named his cat "Puppy" because his kids really wanted a dog but they wouldn't let them and this was the compromise.