I see the changing of his hair color as a physical side effect of him forsaking his mother's nature (his nature), and if you will allow me to speculate, the white hair is symbolic of him embracing his father's nature, not that I think he knows it.
I'm just guessing, but I think Arima (the white hair guy fighting the big Owl, the man Kaneki runs into at the end) is really his father.
They go out of the way in episode 9 of season 2 to show Arima thinking long and hard on what to write in his will, but in the end he leaves it blank. I think it would be messed up to write your child into a will when that child thinks you are dead like, "Dear Kaneki, so I'm not dead, but if you're reading this, then I am dead, so ya, sorry. P.S. Love ya". But the desire to reach out would at that moment be strong. With Arima leaving it blank, he is showing what kind of man he is. A man willing to make the hard choices. The same (more or less) choices Kaneki is making.
At the end, he isn't able to walk away from his friend's request and willingly put himself in harm's way when at that moment no one needed his protection.
That's why I think we see his hair change over and over in the end... a battle between his two natures.