Oooh, interesting question. It seems the answer lies in a combination of cultural standards and the dimensions of our hands.
According to wikipedia, the first keyboards were often based on the tones used in Gregorian chants, so often included a Bflat and a Bnatural on the white keys. So there's your cultural standard. Later, I imagine, they took out the Bflat to make the scale on the white keys Ionian.
Again a cultural standard, but going back to the chants explains why C major was settled on.
Basically some scale has to be on the white keys and it may as well be C major

But why the black keys? Well as someone pointed out they create landmarks to navigate by, but that could be done some other way, however, imagine if all the keys were white .. you wouldn't be able to stretch an octave! And the "sameness" (the double frequency) of an octave is more than just a cultural concept. Cats understand that sameness too. Its fundamental.
Aha! You say, Make the keys narrower! But then our fingers wouldn't fit on the keys!

Haha! I managed to get your list of illogical previous answers into one post! Well done me.

The hexagon "key" board looks very interesting but even if someone had thought of it, I doubt it would have been within the technical capabilities of people 500 years ago.
That will work with an electronic device, where each note is actually a button, but where a hammer has to hit a string that necessarily has to be arranged in parallel rows, it would be very hard to implement.
Oh and .. the hexagon keyboard still represents the same scales that westerners recognise as being "right". There is still no "black" key between B&C and E&F, they are still 1 semitone apart. There's a book called This is Your Brain on Music which explains how much cultural standards in music we absorbed as children just by being exposed to music.
Now lets talk about the qwerty keyboard ....
