What actually happens in "Azura and the Box"?
Solution 1:
A very interesting discussion about this from this Reddit /r/teslore post:
In the Azura and the Box story, the sage Nchylbar was not trying to prove whether Azura could determine what was in the box. He was concerned with whether the divine spirits had mastery over all of the world, or if mortals could create their own destinies. So he duped Azura. She saw clearly what was in the box, but he palmed the flower before revealing it.
In the Aldmer version of the tale, a sphere becomes a flat square, but I'd surmise that the gist of the story still hinges on slight of hand trumping divine power.
Source: http://www.reddit.com/r/teslore/comments/rdbxh/azura_and_the_box_and_halfdwemer_people/c44wcqk
Azura and the Box is one of the Tales of the Ancient Dwemer by Marobar Sul. It is a plagiarisation of a Colovian story adapted to fit a popular conception of the Dwemer. It is not to be regarded as an historical document.
Source: http://www.reddit.com/r/teslore/comments/rdbxh/azura_and_the_box_and_halfdwemer_people/c44y12r
Solution 2:
We might assume that Azura noticed that the flower petal was palmed out of the box, but importantly: the followers in the audience don't know that. So they are led to believe that Azura guessed wrong.
The deception is so bald-faced that we might expect Azura to retaliate, but she had agreed not to harm anyone in advance. The trick of the box is that it isn't the box that's the trick, it's the whole setup: if Azura was all-knowing, she wouldn't have agreed to the encounter in the first place.
Solution 3:
Nchylbar was skeptical of the extension of divine powers.
"Nothing, however, was a greater question to Nchylbar than the limits of divine power. Were the Greater Beings the masters of the entire world, or did the humbler creatures have the strength to forge their own destinies?"
Being the Dwemer 'secretive' by definition it is unlikely you'll find any documentation regarding the exact purpose of his experiment. What we do know, however, is that the 'red flower petal' could have actually been inside the box but removed by sleight of hands.
Solution 4:
I think the most important point of the story is the fact that Azura fled after the box was opened. What seems to be alluded to is that for all their power Daedra can experience the concept of doubt, which for an all knowing, all seeing being, is quite a revealing trait.
This is what I think Nchylbar was trying to prove, not that Azure WAS wrong but that she could be tricked into displaying that she MAY be wrong.
(It feels wrong to refer to Azure as he or it).
EDIT:
Just to add, there is an ongoing (fan based) theory that the power of divine beings in the TES universe is a direct cause of belief. For instance, Talos is a divine because people believe Talos is a divine, therefore Talos is a divine.
Of course, I want to make clear this is pretty much an out of universe observation by fans but stories like Azura and the Box do support the theory. What was it that Azura was afraid of? If a divine being's power is called into question, even slightly, are they really infallible?
Solution 5:
i think it was to show that daedra are capable of being wrong but do not like mortals to know it.
Trying to apply mortal (not only mortal, but human) psychology to such a being is futile, since they are so alien in the way they act and feel.
if she were more human i'd say she were embarassed and angry having just lost face or had known what the dwemer did and couldnt be bothered wasting time on mortals that will troll her.
maybe she went back to moonshadow, to form some vile curse for those involved, as the priest had sensed. The priest was mortified having seen the daedras face before she left, and both the men had to be "carried" out of the chamber by aids, also the head researcher died that night...