"The operation known as hemispherectomy—where half the brain is removed—sounds too radical to ever consider, much less perform. In the last century, however, surgeons have performed it hundreds of times for disorders uncontrollable in any other way. Unbelievably, the surgery has no apparent effect on personality or memory."
Now, according to most views, memory, intelligence, and personality are three areas that either reside completely in the soul, or are affected and changed by it. For example, a person born with a angry temperament can work on himself, and become gentile and kind. This "working on it" is a function of the soul which alters the personality of the person. Another example would be intelligence. At least the function of expressing intelligent ideas through speech ("man became a speaking spirit").
Now of course at some level these parts of our souls connect to the physical world somewhere in the brain, otherwise expressing them would be impossible.
According to this article:
"If the left side of the brain is taken out, "most people have problems with their speech, but it used to be thought that if you took that side out after age two, you'd never talk again, and we've proven that untrue," Freeman says. "The younger a person is when they undergo hemispherectomy, the less disability you have in talking. Where on the right side of the brain speech is transferred to and what it displaces is something nobody has really worked out.""
So in effect, no matter which side of the brain one removes, the person's memory is still there, their power of speech comes back after a while, and their personality stays the same.
Now, if memory is stored all over the brain, then the information in one half would be gone with it's removal. And obviously the brain does not store memory in just the half of the brain that is bound to not be removed! We could suppose that the brain could store duplicate copies of memories in different locations in the brain, but this has never been shown to be true as far as I know.
Also, the ability to express intelligent ideas through speech is a hallmark property of a soul. The fact that it is "relearned" "somehow" on the other side of the brain is not remarkable to those who believe in a soul.
Therefore, I propose that barring any other explanation or experiment, this phenomenon could easily be considered as an additional piece of evidence that the soul is real and interacts with our bodies.
(Not that Torah Jews need such evidence, but it is always nice to see.)