I think we all know the 'real' reason why Khan became 'white' is because the original actors selected decided for one reason or another decided against being in the movie.
I thought the real reason was another vain attempt to keep the secret. "Hey guys, they'll never think it's Khan if we make him a white guy!"
Obviously that's not the reason, or else they wouldn't have originally wanted Benicio del Toro et. al like CommishSleer just mentioned.
They chose Benedict Cumberbatch for the role not because he was a "white guy," but because he was a good actor and he wanted the role. The necessity of the character needing to be dark of skin went out the window once he was cast.
If I wrote a play in which the main character was Irish, but then found out that Denzel Washington wanted to play the part, you better believe I'd be rewriting my play to accommodate him.