Brock the badger is the ideal start, at least for the etymologist, for the animal has been known as a badger since around 1520. Prior to that the largest carnivore in the British Isles was known as a brocc, this an Old English loan word from the languages of the Celtic peoples - Old Irish brocc and Welsh broch two examples. Interesting to note that Middle English brocc was used to mean 'a low, dirty fellow', most often preceded by 'stinking'. Anyone who has ever got close enough to a badger will know they do fit this description, at least aromatically speaking.
Reynard the fox was the title of a series of stories, but the trail goes back much earlier than that. It came to English from Old French in the late 13th or early 14th centuries, the French word came from the Old High German personal name Reginard, itself related to the Latin name Raginohardus - the latter meaning 'strong in counsel' and an early example of the wisdom and cunning associated with this highly successful mammal. The popularity of the stories in France has led to reynard being used as the word for 'fox' in France, replacing the earlier goupil, itself derived from the Latin vulpecula.
Neddy the donkey is only seen in Britain, at least for the most part. In Australia racehorses are known as 'neddy'. Any etymological explanation is lost, but it seems likely to be related to Neddy (or Ned) being used as a pet form of Edward.
Buttercup the cow is probably the obvious answer, butter being associated with dairy and cows. Buttercups were once known by two other names - either a goldcup or a butterflower, thus the modern name borrows a bit of each (and may have been a cupflower). The irony being that certain buttercups contain a chemical called Protoanemonin which is toxic to cattle,
Dobbin the horse was a common name for a working horse, especially a farm horse, and is given as such in Shakespeare's Merchant of Venice, although certainly used earlier. It seems to be a later version of Dob, a pet form of Robin or Robert. Note 'Dobbie', the name of the elf in Harry Potter, was used to mean 'household sprite' from the late 18th century, while in Sussex it had also been used earlier to describe a 'silly old man'.
Fido the dog comes from a Latin word which meant 'to trust or confide in' and most often used to state 'I am faithful', which is what dogs are deemed to be. Personally I think they are only following their natural instincts and if you were to fuss and pet me and feed me whenever I demand, I'd be pretty faithful, too - just don't throw the ball and expect me to retrieve it.
Tiddles the cat is derived from the Old English word tid meaning 'weak, tender'. Indeed, the first syllable of 'tender' shares that origin, while it is still used in English in the word 'tidbit' - yes, I know the English tend to use 'titbit' but the Americans, and quite correctly, have not corrupted it and still say 'tidbit'.
Quackers the duck was the only modern one I could find, and there's no need to explain that, but research revealed that ducks, and indeed many waterfowl, have been known as Arsefoot for more than four centuries. Of course, this refers to their legs being set back on the body (it helps with swimming).
Bobby-Dazzler - now I know you are going to think this is a term used to describe someone of glitzy appearance, but it began as a reference to a butterfly. Indeed, 'bobby' was also used to describe a plant which was covered in invertebrates - buddleia and butterflies, nettles and caterpillars, etc.
Moths were once known as candleflies - no explanation needed there.
Dumbledore - another J K Rowling loan, but it was an Old English word for any buzzing or humming insect.
