Is there something special about how the books
There certainly seems to be a broad pattern about the naming of the books, even if there is not a clear rule of the naming. Robert Arthur linked some of his story names to other books from earlier in the 20th century. For instance:
1. "The Mystery of the Stuttering Parrot" was most likely named after "Poppy Ott and the Stuttering Parrot" by Leo Edwards. In fact, there is even a black mynah bird in that story.
2. "The Mystery of the Whispering Mummy" was most likely named after "Jerry Todd and the Whispering Mummy," also written by Leo Edwards.
3. "The Secret of Skeleton Island" was most likely named after the first book in the famous Ken Holt series, which was also entitled "The Secret of Skeleton Island."
Both the Jerry Todd (Poppy Ott was part of Jerry Todd's world) and Ken Holt books were considered to be high-caliber classic series books; so it's a tribute to them that Robert Arthur used those titles in his own books. Beyond those books, the naming convention went something like this, based on my own observations.
Books were titled "The Secret of..." usually if a clue to the mystery was given in the object mentioned (e.g., "The Secret of the Crooked Cat") or if the mystery took place at the mentioned place (e.g., "The Secret of Terror Castle," "The Secret of Skeleton Island," "The Secret of Shark Reef"). Otherwise, the book began with "The Mystery of the." One exception to this is "The Mystery of Monster Mountain," which could just as easily have been entitled, "The Secret of Monster Mountain."
The words within the title were usually chosen pretty carefully. Usually, a two word description was used so that two words followed either "The Secret of...," "The Secret of the...," or "The Mystery of the..." There are a few exceptions to this, and they really stand out, such as "The Mystery of the Dead Man's Riddle," "The Secret of the Trail of Terror," and "The Mystery of the Creep-Show Crooks."
In titles with "The Mystery of the..." the two words were chosen so that the first word was an adjective of some sort with the second word being a noun, preferably with the adjective indicating some action (e.g., "Stuttering Parrot," "Whispering Mummy," "Talking Skull"). Unfortunately, this sometimes led to ridiculous titles (e.g., "Wandering Caveman," "Dancing Devil," "Cranky Collector"). Another possibility for the adjective was a color of some sort (e.g., "Green Ghost," "Fiery Eye," "Silver Spider"). Of course, where possible, the title was supposed to hint at some unusual mystery that might have supernatural elements; obviously, a mystery entitled "The Mystery of the Runaway Magician" just does not sound as ominous as "The Mystery of the Talking Skull."
Sometimes the titles were a little misleading or just not clear enough. I remember as a child reading "The Mystery of the Vanishing Treasure" and not having a clue as to what the title was referring. Also, "The Mystery of the Shrinking House" was about a series of paintings, not about a house that shrinks. And the title construction led to some silly titles at times, such as "The Mystery of the Sinister Scarecrow." Overall, though, the titles were a cut above titles in other series books both then and now, and it's great to look at the titles even today and see what images they evoke in the mind's eye.