I'm sure you're thinking the name is popular from the movies and/or books you were exposed to when you were younger. After reading Shilo, and Where the Red Fern Grows as a kid, I went on a spree of reading any book I could find with a similar premise, and I noticed the same thing you did, that there was only a small number of names that get used for dogs.
I think it's a common theme in books and movies to name the dogs something simple, so that it's not only easy to remember, but it's also easier to distinguish between human and non-human characters when the non-human characters don't have human names. In music, I think dogs gett named whatever is easiest to rhyme with, or just something that sounds good. I think the name Blue is a good name for a country song because of how the word sounds.
The majority of the stories that I was exposed to took place in the American countryside, so it makes sense that the dogs were dogs that you would typically find in the country. The typical dog you're going to have in the country is going to be one of the most common hounds in America, the Coonhound. Coonhounds come in two "colors", red and blue, so when you hear a name like "Blue" it's most likely referring to the color of the hound.
(It might be worth mentioning that there are other breeds of dogs that are "blue", and that they also commonly get named Blue as well. There are Blue Heelers, Blue Bulldogs, and Blue Terriers to name a few.)
Note: Click on the images for more information
The American English Coonhound aka the Redtick Coonhound:
These hounds were brought to the Americas in the 17th century and are pretty common in the south I think. One of their nicknames is the Virginia Hound.
You can tell how they get their name, as they have a distinct red and "white" pattern, with red ticks.
The Redbone Coonhound:
Imported into America not long after the Redtick Hound. The Redbone hound comes from a selective breeding of Foxhounds and Bloodhounds in order to be able to hunt tree-climbing prey, such as raccoons.
This is the hound referred to in:
- Where The Red Fern Grows
- VR Troopers
- The Hound That Thought He Was a Raccoon
- The Fox and the Hound (Novel)
The bluetick used to be classified under the foxhound, but was recognized as it's own breed in 1946. You'll notice that the markings are very similar to the redtick hound, but the markings are a dark black, and the tick marks are a dark blue. The blue color is quite often only noticeable as a sheen in the sunlight, so pictures here won't do it justice, but it's that blue color that gives them their name.
This is the hound that is referred to in:
- Where the Red Fern Grows (Though they weren't the main hounds)
- The song Old King by Neil Young is about his Bluetick hound he had
- The song Red Dirt Girl by Emmylou Harris
- The song These Old Bones by Dolly Parton
- The movie Wild River which he was actually called "Old Blue"
- One Flew Over the Cuckoos Nest
- Savage Sam (The sequel to Old Yeller)
- Summer of Monkeys(Novel)
As you can see, there are a lot of references to blueticks, and I don't know how many of them are actually named Blue, but it's probably safe to bet that there are enough of them to leave an impression.
Now, you also mentioned the bloodhound. There is also a common inference with naming animals based on the sounds they make. In this case, nearly every hound will howl, and in some hounds like bloodhounds, it will be a howl similar to "balooo".
Another option, which I think is less-likely, bloodhounds used to sometimes referred to as blue-blooded, which stems from older references to the breed as a blooded hound, meaning they're of pure breeding stock.
The only bloodhound I can think of called Blue, was in the Andy Griffith Show when Barney Fife thought that a sheriff's office needed a tracking dog and hilarity ensued.