Having just finished re-watching David Tennant's run on Doctor Who I was suddenly struck by a theory as to who John Hurt's Doctor really is. Now, this isn't ego-driven, but I'll do it anyway: potential spoilers ahead.
My theory starts in "Waters of Mars", one of the three "specials" before "The End of Time". In it, The Doctor goes on a miniature power trip, claiming to be 'Timelord Victorious', declaring that the power of time should bend itself to him. You may notice that I've captialised the word 'victorious' there. I'll come back to that.
Next, in Tennant's last episodes, "The End of Time" parts one and two, the three primary Timelords involved are referred to as "My Lord..." (Doctor, Master and President).
My theory, therefore, is that John Hurt is actually... the Timelord Victorious. A parallel 11th Regeneration that Tennant's Doctor would have become had he gone down the route of power over time, rather than abandoning it at the end of "Waters of Mars". I think that moment, where he loses himself, created a schism in time - an alternate Doctor - that would eventually regenerate into John Hurt, aka Victorious.
It's his new name - and with the power of time at his control is able to infiltrate our 11th Doctor's reality. I also think that's why Tennant's Doctor is in it - as he is the source of the schism and the solution to fixing it.
I'll finish with the exchange at the end of "The Name of the Doctor":
Doctor: I said he was me, I never said he was the Doctor.
Clara: I don’t understand.
Doctor: My name, my real name, that is not the point. The name I chose is, “the Doctor.” The name you choose, it’s like a promise you make. He’s the one who broke the promise. He is my secret.
Other Doctor: What I did, I did without choice.
Doctor: I know.
Other Doctor: In the name of peace and sanity.
Doctor: But not in the name of the Doctor!