I think higher brain power is just another way of expressing the way we tend to everyday situations...to be honest, this is a very opinionated question with many answers...an academician believes it to be the ability to study or excel in academics, an artistic person may believe in the creativity of it, an athletic person may see it as strategic knowledge levels, a religious person may believe it to be divine, etc.
But I believe in this:
A disabled person may believe it to be the ability to function normally...this is the truest answer...the ability to function in everyday life in everyday situations with more ease than other organisms, or as we strangely call them, 'lower life forms'...that, itself, is to be the reason we can call ourselves a species of 'higher brain power'..