But if you have a character who goes by the name "Khan" he should be a brown man, most definitely. Khan has its historic roots in the real world as a title from the Mongols and successor states, as well as in Trek-lore. If there was a character called Pharaoh but they cast a Chinese actor I would also present a similar unease. Just like I wouldn't accept a Black Godfather because the Godfather should be an Italian or someone who can convincingly play and Italian.
Not all parts are the same, and for some of them the description includes a certain race or ethnicity. Benicio del Toro would be better cast for the part than Benedict Cumberbatch, but like I said I would go for someone ever browner like Naveen Andrews, because Ricardo Montalban was yes Latino, but as a Mexican I am guessing Mestizo and also a brown man, where as Benicio is more olive, but Cumberbatch is a pastry Brit and that doesn't work, even though he is a fine actor in his own right.
Benicio del Toro would add the menace Cumberbatch lacked; I've seen him as villains in other movies and while it sounds like typecasting (okay, it sorta is), he would have been a better choice. That was my only problem. Benedict's half-choked gnawing of "I am Khhhhaaaannn." almost had me laughing in the theater and, come on, he's a great actor as well - which only made the situation worse. (To compare, Spock aping the infamous "KHAAAAAAAAN!" line certainly had me rolling over how amateurish the flick was at times, as it reminded me of something I wrote for creative writing back in 9th grade a few decades ago...)
Skin color didn't faze me as such (people of mixed skin colors exist so someone with a rich ancestry from the Asian region can still look pasty white and, by DNA, be relevant. Or if a person's lineage is 100% pure-bred Irish, I don't see what the problem necessarily is. As long as one understands the heritage behind the name itself.
What's funny is, "Khan" is usually a title like "President" ore "Prime Minister". Or "Chieftan via heredity", et cetera, et cetera. It stems from the Mongolian region.
"Singh" is Indian in origin.
Right away, there's a mismatch but I'll get back that in a moment.
"Nooinien" has no history of which I could find within a 5 minute span of time. So it is potentially nothing more than fiction that sounds like an authentic name from a region in the "East", and am using quotes because "East" and "West" are surely also supposed to be passe in a globalized world, which brings us full circle:
So given the hodge-podge accorded the name "Khan Noonien Singh", and how many Indian people are technically Caucasoid at a genetic level, not to mention Australoid, Mongoloid, Negroid, and other classifications of humans--
Forget the fact there is only the human race, which contains many subtle differences (even within a classification-- I am probably wrong or incomplete in that assessment, certainly for just one paragraph, but I just don't see "____oid" as being a race but a feature of the human race) but they don't make people into wholly different species as a result...
Even Montalban had dark make-up applied. At least in 1966. In 1982, he did not and looked as white as anybody else. Didn't stop people from being affected by his performance... so that's already proof that skin color made no difference.
So is there no reason a white person can't be given the name, apart from "only brown people can have the name"? The question, I'd opine, is the efficacy of writing and acting, for which "Into Darkness" was hit or miss and while the retooling of Khan's origins was not bad, there's no reason for Khan to be there. There's plenty going on without the fanservice and the need for "magic blood" is nothing approaching a valid reason.