I am highly agnostic with respect to reincarnation, but there is some evidence for it:

1. Peer reviewed research done via investigation of past-life reports suggests that it is indeed possible for a child to be born with accurate memories that are not his own, and that could not be obtained in any other way.

2. While the processes of thought and memory are, to some extent, understood by neuroscience, consciousness itself is still a complete mystery, in science as well as philosophy.

3. This is not strong evidence, but it counts for something: many cultures and religions have stories describing reincarnation. This idea is described in great detail in Hinduism, Judaism, and Buddhism.

But of course, until we understand the mechanism used in reincarnation, it will remain in the domain of myth.