- DNA, since it "codes" much of who we are here on Earth. This answer sounds fine on the surface (and indeed, may be a helpful indicator of when life begins), but it's pretty flawed, scientifically speaking. Mutagens are substances which cause DNA mutations. In other words, if your precise DNA code is what makes you "you," then "you with cancer" is a separate person whose "birthday," so to speak, is different from your own.
- Your personality. The best argument for this position would be Phineas Gage, a railroad crew foreman who, after suffering traumatic brain damage (from a chunk of hot iron being shot through his skull by mistake), became what might be called "a new man." As his physician put it:
His contractors, who regarded him as the most efficient and capable foreman in their employ previous to his injury, considered the change in his mind somarked that they could not give him his place again. He is fitful, irreverent, indulging at times in the grossest profanity (which was not previously his custom), manifesting but little deference for his fellows, impatient of restraint of advice when it conflicts with his desires, at times pertinaciously obstinent, yet capricious and vacillating, devising many plans of future operation, which are no sooner arranged than they are abandoned in turn for others appearing more feasible. In this regard, his mind was radically changed, so decidedly that his friends and acquaintances said he was “no longer Gage.”
The current theory about what lead to Gage's transformation is that the accident functioned as a makeshift lobotomy that damaged his frontal frontal cortex, removing his social inhibitions. In a way, this 'new Gage' was perhaps lurking in the personality of 'old Gage.' This cannot be the defining standard that creates the "essence" of a man. Where would such a standard end? Truama of all sorts, physical or otherwise, can fundamentally alter a person's character. So can things like education, a new-found faith, and so forth. And what of temporary changes to personality? After all, a man who imbibes a bit too much on alcoholic beverages may find himself acting "not quite himself," but this is merely a figure of speech. Should we really claim that before four drinks, you're yourself, and after, you're a "different person" who is born and dies with the cycle of your sobriety?
- Your memory. This one is related, but not identical, with personality. As bizarre as Gage's case was (the guy started to walk off being shot through the brain with a chunk of iron), he still had a memory of who he was before. But what if you woke up one morning like Jason Bourne from the Bourne Trilogy, with no recollection of who you were prior to that day? Of course, this standard is as problematic as all the rest. What of people with amnesia or Alzheimer's, who chronically forget? Do we claim, "oh, you're the twentieth in a line of people named Mary to inhabit your body"? Or, you're not a person at all, because you can't remember anything? Would two people with equally blank memories be considered the same person? Obviously, this standard won't do, either.
- Your body. Going in a different direction, one might speculate that the physical boundaries of a person, so to speak, define who they are. Some perverted form of this seems to arise in the abortion debate. The unborn child, with a discreet and identifiable physical shape and form, as well as a distinct brain, organs, and DNA, is considered by law to be the same "person" as the mother, since they inhabit the same body. But immediately, there are problems: conjoined twins share a body, but are two people; people are amputated, give and recieve blood, or have organ transplants, and remain "themselves." So body, then, is clearly out.
- Your brain / mind. But what if, instead of defining a person by their body, you identified them just by their brain or mind? For the physical organ, the brain, there are some theoretical problems. There's a hypothetical procedure called either a "brain transplant" or a "whole body transplant" (since theorists couldn't decide if the body was getting a new brain, or the brain a new body). There are also really serious questions, due to hypothetical "partial-brain transplants," where a chunk of one person's brain is fused with someone else's. A science-fiction book, My Brother's Keeper, by Charles Sheffield, apparently examines this issue by having two twins (a concert pianist and a spy) in a terrible accident undergo surgery. The patient who awakes has the personality of the pianist, but the memories of both, so of course, he undergoes some secret mission. Would the patient, post-procedure, be considered the pianist, the spy, or a new person completely?
Still, of all of these possibilites, the brain, and more specifically, the parts of the brain we call the "mind," which help create self-awareness and self-identity, serves as the most accurate answer. But it's only partially correct. From a theistic perspective, the answer is easy enough. The "soul" is what we consider the defining characteristic of a person. It's invisible, distinct from the physical phenomenon, and yet interacts through them. It's impossible to transplant (even theoretically), and completely indivisible, so it's immune from that reverse Sorites' paradox. It's impossible to say the precise interconnection between the soul and the mind, because we don't know much about the way that the brain works, and we'll never know the depths of how the soul works. So to the extent that the mind seems to be the primary playground of the soul, and the hotspot for temptations to the soul, I think brain/mind is at least a half-right answer for what makes us "us."
One final piece of the puzzle. In Mark 7:14-23, Jesus identifies the "heart" as being the source of all sin - that it is not the physical things we come in contact with, but how we decide to react to them that determines sin. This was, for example, why the early Christians declared the raped virgins of Rome to still be virgins. What He means by "heart" is easier understood than defined: it's what any of us mean when we say someone "has a lot of heart," or "sang with all their heart," and it pretty definitively doesn't refer (at least primarily) to the physical organ.
So heart, soul, and mind... where have we seen that trinity before? Oh right. Jesus already suggested this answer in Matthew 22:37.
So the soul, working through the heart and mind, defines the essence of an individual. But what about the essence of the Holy Trinity? What is the defining characteristic of God? I'm going to try and tackle that one tomorrow.