Going with the beard theme, Rasputin perhaps? Shortens nicely to Raspy or Putty.
My mom always wanted to name a cat “Leonard Finklestein, Esq.” At one point, my sister had a cat named “Biff the Einstein Cat,” mainly because it was the stupidest feline ever to run up my sister’s curtains.
Like many other Dopers here, I’d go with naming her for the calico thing, but I don’t think I’d go with some variant of “calico” or “patches” for it. There are other words you can play with. How about “Splochisplotch,” which is what I’d probably end up calling her. She’s colored as if she’d splashed around in a puddle or two. So maybe “Puddles,” or “Puddly,” or “Puddlepuddle” would work well. If you’re feeling grand, why not “Puddleoupe, Queen of the Caribbean.”
Or maybe you could go with an anti-cute name, like “fuzzface” or “lintball.”