Let's not look at other avenues for increasing intelligence in an animal of a given size. What changes to known biology are necessary to allow individual nerve cells to be significantly smaller and consume less energy without reducing their processing power? I know insect neurons are (at least sometimes) much smaller than human neurons, but I recall reading that those tiny insect neurons have far fewer dendrites. I vaguely recall calculating that the volumetric density of dendrites was almost the same in insects and humans.
I've seen What is the smallest sentient being possible? (and its linked question), Can we scale down humans to the size of a mouse? and Scientific Accuracy - How can we make an alien brain more 'efficient' than a human's? The first two questions and their answers aren't focused enough, and even in the third, only one of the answers starts to get at my specific question. Note that making neurons faster is one method of increasing a brain's processing power, but that in itself is not an answer to my question. Being faster doesn't let you do all the things having more components does; most obviously, it does nothing to increase memory capacity.
Clarification: I meant "change" in the sense of "difference from Earth biology" rather than "modification." My main question isn't "How to engineer Earth humans or animals like this?" (though that would be an interesting bonus) but "What naturally-occuring alien biology could be like this?" That said, alien biology can be extremely different, and how close can you get to Earth biology while achieving this?