First, the name couldn't be gibberish. Naming your dog "Asjbafbawlb" sure would be quirky, but it's also unpronounceable. That's just being strange for strange's sake.
Secondly, the name couldn't just be from another language. There are probably areas in the world where those names are common, and thus are common dog names.
Thirdly, names from mythology had to be limited. It would be so easy to fill a list with only names of fantastic beasts of yore! It would be a great list, but not show the breadth of possibility out there.
This list should at least get you thinking about the kind of names that are going to stand out. Do you want your pup to be the third "Lucky" at the doggie day care center? Do you want to avoid calling over a whole pack of mutts when you tell your pup it's time to leave the dog park? Read this list and follow the themes to come up with the pawfect, unusual name for your puppy.