First, please consider this syllogism:
Patterns recur in nature.
Awareness occurs in patterns.
Therefore awareness recurs in nature.
A baby’s awareness could develop in utero in such a way that it had the same (or very similar) awareness as a baby from 1,000 years prior. Thus, it would be the same baby, since awareness is identity. Awareness is identity because awareness is the way a person processes pleasure and pain. As long as all the essential parts of a baby's awareness were functioning with respect to a previously-existing baby the two babies could have the same awareness. By analogy, a song may be played with a lot of reverb or with no reverb but it could be the same song in essence because it has the same notes.
Something like intelligence is not an aspect of the mind that feels pleasure or pain, so it is not the essential identity of a person. No matter how much a person changes he can only become himself.
The brain processes information and stimuli in predictable ways, in patterns. The awareness of sadness is the same for me as it is for you. Otherwise empathy would not be possible. The same parts of the brain are responsible for self-awareness in every person, so that self-awareness is one thing, one pattern of activity. But more importantly, it is the same feeling for everyone.
Obviously twins aren't the same person, and if one twin dies he doesn't jump into the other twin's body, because a body can only house one person's consciousness (usually, I think). I'm not talking about anything spooky here. I'm saying that if a baby is born who is exactly like or very similar to a baby from a thousand years ago (with respect to the mind) those babies are one and the same. Furthermore, it would only be necessary that the baby that is born today have the same cognitive starting point (that is, the same identity) as the baby from 1,000 years ago. The two babies could have different genomes and divergent life paths.
While it is not possible for for an organ, say the liver, to experience a second life, it is possible for patterns of awareness to be reborn or to recur in nature, because patterns of awareness are not, themselves, physical things, but impressions made by physical things. It is possible for me to think the same thought someone else first thought. Thoughts are impressions made by neural activity. But thoughts are not the ultimate basis for awareness. Rather, the neutral feeling one returns to after being emotionally excited--the plain awareness which one feels while in a somewhat relaxed state--is the basis for reincarnation, because that state is common to all people who are not in a neurotic or psychotic state. It is this commonality which is the basis for reincarnation. As long as a baby born today has the same starting point in common awareness as a baby born 1,000 years ago, the two babies would have the same self or identity. Because of this I am not a strict materialist.
Dr. Oliver Sacks said that when a person hallucinated an image of teeth that hallucination occurs in a certain part of the brain. Likewise different parts of the brain store different images. The brain has a certain generalizable way of processing different bits of information in different parts of the brain and in different predictable ways. These different processes unfold as patterns and are felt in specified ways. This is what I mean when I say that awareness--perceiving pleasure and pain--is patterned. In fact, the brain itself is laid out in a patterned way. Though the brain itself, as a physical organ, is not what reincarnates, but the awareness. Awareness is a method of perceiving, not a physical organ. Emotions have certain ways of being felt--anger, sadness, joy. We recognize them and group them according to their familiarity. So in this sense also, awareness is patterned.
What I’m not saying: I’m not saying that a soul or immaterial essence is necessary to produce this effect, namely reincarnation, though I can’t rule out a soul. I’m saying that this effect is the necessary result of the way awareness develops in utero and perhaps in early infancy, that the patterns of brain activity that cause awareness to emerge simply repeat across lifetimes.