I believe the line should actually be, "Bells on bobtail ring," or "Bells on bobbed tail ring," rather than 'Bobtail' being the proper name of the horse. He isn't named 'Misfortune' either.
Many sites such as the one above have, "Bells on bob tails ring," but as it is a one horse sleigh, there is only one tail. Bobtail should be one word. If you're going to make it two words then it should probably be "bobbed tail."
Bobtail, as one word, is used as a noun to refer to both a bobbed tail or the animal with a bobbed tail. It is also used as an adjective to describe an animal with a bobbed tail. If I'm not mistaken, the tails were often bobbed (cropped or docked) on horses pulling vehicles to keep them from becoming entangled with the reins and brake.
Main Entry: bob·tail
Function: noun
Date: 1605
1 a: a bobbed tail b: a horse, dog, or cat with a bobbed or very short tail; especially : old english sheepdog
2: something curtailed
— bobtail or bob·tailed adjective
http://www.m-w.com/dictionary/bobtail
