I'm guessing that the answer to "The Question" will reveal a number of big secrets.
The first secret is that his name, per se, is not the answer to the question, nor what the question is really asking for in the first place. The question is asking "who" the Doctor is on a much deeper level than that of a mere name. A mere moniker is pretty meaningless devoid of a meaningful context, isn't it?
The second secret, is that while the Doctor will answer the question truthfully, as the prophecy states, his answer will likely result in one of those situations where this new info merely presents us with even more questions than we started with. The truthful answer won't
solve any great riddle about the Doctor, but merely establish that an even greater riddle exists, and one which he has been totally unaware of until forced to answer the question of "who?"
And the quest to discover the nature of this new riddle, and it's mysterious answer, will simply provide an added element to the Doctor's adventures over the next 50 years or so of storytelling. (I'm guessing that his truthful answer to "Doctor Who?" will be, "I don't know". And having to finally face the fact that he doesn't really know "who" the hell he is, or what really motivates him, sparks an open-ended quest to discover the secrets of his own past/place in the universe/etc.)