Nautilus has an article[1] that explores exactly this question. I beleive the conclusion was that it is the number of cortical neurons, not the total number of neurons and not brain mass that best predicts intelligence, or at least best correlates with our intuitive ranking of intelligence that puts humans and then primates at the top, above elephants.
No, an elephant needs a larger brain to run its larger body. Humans have enough brains to run our bodies, plus "extra" with which to think, in a manner of speaking.
I've always thought the metabolic needs must have been just as driving (specifically, being carnivores). Note: not a dig at vegetarians, we're talking about long before the proliferation of tofu and quinoa.
"A gram of brain tissue takes 20 times more energy to grow and maintain than a gram of tissue from the kidney, heart, or liver"
