Interesting question, Illiadus!
I like this sort of discussion very much.
You said the magic word yourself in your question. The evidence must be
objective. Above all else, the evidence must be objective & falsifiable.
Hypnotism, for example, is most certainly not objective. It is a highly subjective process that has been shown capable of creating false memories. Therefore, hypnotism results should not even be considered as proof of reincarnation.
Similarly, personal accounts of "impossible memories" from children and/or adults should not be taken as proof, because of the highly subjective nature of memory and the collection methods of this sort of data.
Physical attributes such as birthmarks in the same location as known wounds of the supposed reincarnated person, or a facial resemblence are too easily accounted for by natural explanations should not be considered as meaningful.
As with everything, many pieces of bad evidence do not equal one piece of good evidence. A man with who claims he was hung as a criminal in a past life, who has a birthmark that looks like a ropeburn on his neck, who is terribly uncomfortable around policemen, and has recalled many accurate details of a past time both under hypnosis and while awake, has not proved his claim. There are better, natural explanations for all his evidences. He has not proved it one bit more than if I simply said I believed I was reincarnated because I love medieval history. His story is a lot more interesting, however.
It is much easier to describe faulty evidence for reincarnation than to imagine what sort of evidence would be sound enough to prove it. Since all evidence worth considering is based on the feelings or knowledge the the person making the claim, what would be required is for them to come forth with some sort of falsifiable statement about there past life. That would be very difficult if not impossible to do. The statement would have to be specific and unique enough to exclude the possibility of a chance "hit" or guess.
I see that I have spent more time talking about bad evidence than the valid sort here. Perhaps that is understandable in this case. Part of the big problem with proving reincaration is that there are so many "flavors" of it. Everyone that believes in it has a subjective idea of how it works and what its purpose is, but these are metaphysical concepts. before we can determine the truth, it would have to be studied. Before we can study it, it would have to be proven at the most basic premise: that a human conciousness can be transfered, perhaps many years later, from one body to another. That is a tall order.
Regards, Canis