Two things about this:
1) The AI would have to surprise us not just about the fact, but all observations therewith entangled. Eliezer_Yudkowsky mentioned in one comment the possibility of it telling us that humans have tails. Well, that sounds to me like a "dragon in the garage" scenario. What observation does this imply? Does the tail have mass and take up space? Is its blood flow connected to the rest of me? Does it hurt to cut it off?
2) For that reason, any surprise it tells us would have to be sufficiently disentangled from the rest of our observations. For example, imagine telling someone ALL of the steps needed to build a nuclear bomb in the year 1800, starting from technology that educated people already understand. That is how a surprise would have to seem, because people then weren't yet capable of making observations that are obviously entangled with atomic science. Whether or not the design worked, they would have no way of knowing.
So an answer to this question would have to appear to us as a "cheat code": something that you have to make a very unusual set of measurements (broadly defined) in order to notice. On that basis, one answer I would give to the question would be the "cognitive blind spot" common to all humans that can be exploited to make them do whatever you tell them. And that method would have to be something that people would never dream of doing. Not just "hey that would be morally wrong", but "huh? That couldn't work!"
Imagine something like those "hypnosis terrorists" that trick random people into giving them stuff, but much weirder, much more effective, and which results in the victims feeling good about whatever they were tricked into, all the rest of their lives, and showing all signs of happiness on all MRIs and future brainscan technologies when thinking about their acts. (I'll post a link about hypnosis terrorists when I get a chance.)
Does the tail have mass and take up space? Is its blood flow connected to the rest of me? Does it hurt to cut it off?
Well, yeah. Obviously.
Human beings are all crazy. And if you tap on our brains just a little, we get so crazy that even other humans notice. Anosognosics are one of my favorite examples of this; people with right-hemisphere damage whose left arms become paralyzed, and who deny that their left arms are paralyzed, coming up with excuses whenever they're asked why they can't move their arms.
A truly wonderful form of brain damage - it disables your ability to notice or accept the brain damage. If you're told outright that your arm is paralyzed, you'll deny it. All the marvelous excuse-generating rationalization faculties of the brain will be mobilized to mask the damage from your own sight. As Yvain summarized:
I find it disturbing that the brain has such a simple macro for absolute denial that it can be invoked as a side effect of paralysis. That a single whack on the brain can both disable a left-side motor function, and disable our ability to recognize or accept the disability. Other forms of brain damage also seem to both cause insanity and disallow recognition of that insanity - for example, when people insist that their friends have been replaced by exact duplicates after damage to face-recognizing areas.
And it really makes you wonder...
...what if we all have some form of brain damage in common, so that none of us notice some simple and obvious fact? As blatant, perhaps, as our left arms being paralyzed? Every time this fact intrudes into our universe, we come up with some ridiculous excuse to dismiss it - as ridiculous as "It's my daughter's arm" - only there's no sane doctor watching to pursue the argument any further. (Would we all come up with the same excuse?)
If the "absolute denial macro" is that simple, and invoked that easily...
Now, suppose you built an AI. You wrote the source code yourself, and so far as you can tell by inspecting the AI's thought processes, it has no equivalent of the "absolute denial macro" - there's no point damage that could inflict on it the equivalent of anosognosia. It has redundant differently-architected systems, defending in depth against cognitive errors. If one system makes a mistake, two others will catch it. The AI has no functionality at all for deliberate rationalization, let alone the doublethink and denial-of-denial that characterizes anosognosics or humans thinking about politics. Inspecting the AI's thought processes seems to show that, in accordance with your design, the AI has no intention to deceive you, and an explicit goal of telling you the truth. And in your experience so far, the AI has been, inhumanly, well-calibrated; the AI has assigned 99% certainty on a couple of hundred occasions, and been wrong exactly twice that you know of.
Arguably, you now have far better reason to trust what the AI says to you, than to trust your own thoughts.
And now the AI tells you that it's 99.9% sure - having seen it with its own cameras, and confirmed from a hundred other sources - even though (it thinks) the human brain is built to invoke the absolute denial macro on it - that...
...what?
What's the craziest thing the AI could tell you, such that you would be willing to believe that the AI was the sane one?
(Some of my own answers appear in the comments.)