It's hard to argue for a claim (that seems to me) so patently obvious. But I'll try. I take it we want our concept of a 'person' to highlight what is most important about a life, what gives that creature an intrinsic moral worth. Now it's surely clear that some form of mental life is absolutely essential to moral worth. Animals may not be persons, but we might reasonably have concern for their interests insofar as they can suffer or enjoy life. Plants or rocks, by contrast, warrant no such concern. So we should consider mental life to be a necessary, if not sufficient, feature of a person.
The alternative is to claim that any human organism is, ipso facto, a person. On this view, what really matters is having the right biological (rather than psychological) makeup. As Chris rightly notes, this view is insulting to the dignity of persons: "It results in the view that all of the things that make us different from one another -- all that makes us who we are -- is superfluous, merely icing on the cake of undifferentiated vital functioning."
To quote David Velleman:
Now, I don't believe in the existence of a soul conceived as a spiritual entity, a piece of spirit-stuff that inhabits my body. But there is a sense in which I believe everything that is most important -- and, in I think, everything that is morally important -- in the doctrine of the soul. For I believe that I am not just my body: I am something more than my body, something that could in principle survive the death of my body (though I do not believe that it actually will).
I don't call this "something more" a soul; I call it a mind. But in some languages -- ancient Greek, for example -- the word for mind (psyche) was also the word for soul. And just as others regard the soul as the basis for the moral status of persons, so I regard the mind.
I think that there is an important difference between opposing all abortion on the grounds that the soul is present from conception and opposing it on the grounds that the fetus is human. I disagree with the former view but I understand and admire it. The latter view strikes me as the sort of thing that is usually charged against secular academics: soulless materialism.
Now, Brandon complains that this restrictive view of personhood is unhelpful, since it doesn't tell us how to treat non-person humans:
We cannot apply our commonplaces about corpses, despite the oxymorons I've read (e.g., "animate corpses"); if we were dealing with corpses, properly speaking, no issues would arise, because the sort of questions we have to ask about those who have experienced higher-brain death cannot arise in the case of corpses. You do not fret about whether a corpse should die; if you can rationally do so, you are not dealing with a corpse.
But I think the view does provide at least some guidance. After all, since the person is already dead, we aren't fretting about whether they should die, but merely their body. It would seem reasonable, on this view, to treat their bodily remains as we normally do, i.e. in accordance with the wishes of the body's deceased owner. If a person wishes their body to be removed from life support once the person is gone, then that is exactly what we should do. There's nothing to "fret" about. To sustain the body in ghoulish animation against the deceased person's wishes, is as repugnant as burying them when they expressed a preference for cremation. It's just plain disrespectful.