To answer the point about Ian Stevenson: I do stand by my opinion of Stevenson's research. The 'rebuttals' that are often mentioned are simply denials, based on the firm conviction that his findings must be false, or he must have been wrong, because we know
that there can be no basis to apparent past-life memories. But to quote just one of his documented cases:
In Sri Lanka, a toddler one day overheard her mother mentioning the name of an obscure town (“Kataragama”) that the girl had never been to. The girl informed the mother that she drowned there when her “dumb” (mentally challenged) brother pushed her in the river, that she had a bald father named “Herath” who sold flowers in a market near the Buddhist stupa, that she lived in a house that had a glass window in the roof (a skylight), dogs in the backyard that were tied up and fed meat, that the house was next door to a big Hindu temple, outside of which people smashed coconuts on the ground. Stevenson was able to confirm that there was, indeed, a flower vendor in Kataragama who ran a stall near the Buddhist stupa whose two-year-old daughter had drowned in the river while the girl played with her mentally challenged brother. The man lived in a house where the neighbors threw meat to dogs tied up in their backyard, and it was adjacent to the main temple where devotees practiced a religious ritual of smashing coconuts on the ground. The little girl did get a few items wrong, however. For instance, the dead girl’s dad wasn’t bald (but her grandfather and uncle were) and his name wasn’t “Herath”—that was the name, rather, of the dead girl’s cousin. Otherwise, 27 of the 30 idiosyncratic, verifiable statements she made panned out. The two families never met, nor did they have any friends, coworkers, or other acquaintances in common, so if you take it all at face value, the details couldn’t have been acquired in any obvious way.1
Stevenson documented many such cases, comparing each detail in the child's story with witness testimony, newspaper reports, interviews, and the like. During this, we observed methods that even his many doubters agreed constituted proper scientific protocol, and gathered, over 30 years or so, some thousands of such cases.
One of the intriguing aspects of his many stories was that children would often display birthmarks in proximity to the site of injuries which had resulted in the death of the previous 'incarnation'.
I don't think one ought to be overly fascinated by these accounts, but if asked 'what possible evidence for mind existing apart from the body', then obviously such accounts are one of the few sources of empirical data
that you can point to.
As to the contention that this is more like the Hindu belief in 're-incarnation' - well, here you enter difficult territory. But consider that in Tibet, for instance, the succession of incarnate lamas is determined by (among other things) the ability of the child to recognize the possessions of the former incarnation. In Buddhist terminology, the use of the word 'soul' is generally frowned upon, for doctrinal reasons; but 'mindstream' is regarded as quite acceptable. And I think it serves the purposes of this point perfectly well.