the initial role of big brains was for sexual attractiveness. For a long time, hominid females were selecting for big brains. Big brains served the same purpose as big tails do for peacocks. I think perhaps that those females were determined to someday produce a man with the mental capacity to understand women.
While this abstract is slightly funny, I suddenly realized that I have no idea about why humans evolved big brains. Why most of the other species do not select for bigger brains, while human evolution seems to suggest that a bigger brain is a much better survival tool compared to sharp claws or protective colour? Certainly, other species would have offspring which have slightly larger/smarter brains. Why is that only a few species (humans, dolphins ...) evolved relatively large brains?