How many times have you heard someone say that as an insult? It's meant as a disparaging remark, implying that even half a brain would make someone smarter than they appear to be at that moment. The same people who use that expression, would be surprised at what half a brain can do.
The human brain has two halves, often likened to the shape and nature of half a walnut. Cells within that brain are organized in a way that is particular to humans and primates, that is, having the centers that control higher levels of thinking, such as organization, located at the front of both lobes.
On the other hand, the dolphin's brain while having the two lobes, divides certain basic functions to each lobe, an ability that likely keeps it alive. Because dolphins must "breathe" constantly, only one half of their brain "sleeps" at any given time, allowing the other half to run their respiratory system.
Dolphins have a large brain, relative to the size of their bodies, but it appears to be the structure that contributes to their unique intelligence. Scientists studying fossil remains of the dolphin's ancestors, have discovered that their skulls underwent the same type of encephalization as humans and primates, when the bone structure changed to accommodate a brain that grew in size, as it developed in thinking and reasoning capacity.