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I think that SeijiSensei is on the right track.
In a true Mystery show, the mystery drives the plot. The show is either about one specific question (like the ones that SeijiSensei listed) or if the show is more episodic, each episode is centered around its own particular question.
With this in mind, I think that a really good, recent example of a Mystery show was Un-Go. In the early going, each episode centered around a mystery, typically of a "Whodunnit?" nature. Later on, you had story-arcs that spread across multiple episodes, with each arc dedicated to certain mysteries.
The mysteries of Un-Go were arguably of variable quality (in the sense of how difficult they were for the viewer to figure out), but the mystery was consistently "front and center" in the content of each episode. Sure, it was still a character-driven show in many respects, but many of the key characters themselves had an inherent passion for solving mysteries.
I've only seen the first two episodes of Hyouka, but my impression of it so far is that it approaches its mysteries as nice side-attractions to spice up the show a bit, and also perhaps to differentiate it some from other "school life" anime (I use the term "school life" for lack of a better term). There's nothing necessarily wrong with Hyouka's approach, but in a true Mystery show, the Mystery is front and center. You can tell this by how much time the characters spend focusing on the main mystery of the episode, and by how engaged they are by it.
In Un-Go, the mystery of the episode would take up the majority of the episode, if not virtually all of the episode in some cases. In Hyouka's first two episodes, the mysteries might take up a few minutes, if I had to guess.
Now, maybe Hyouka is just "warming up" here, so to speak, and will become more mystery-focused the deeper you go into it.
One final point - Keep in mind that a show can really be a true Mystery show, but also simply be bad at how it executes its mystery element. Deus Ex Machinas can indeed be a huge problem in mystery shows. While Mystery fans are used to their brilliant Sherlock Holmes-esque Detectives who cleverly deduce/uncover the truth, it shouldn't feel too magical. It's a thin line that can be hard to see, but it's there.
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