Homo sapiens and Neanderthals overlapped in Europe and western Asia for somewhere between 2,600 and 5,400 years, depending on which dating you trust. During that period they shared space, interbred occasionally, and presumably had some kind of contact. Then the Neanderthals disappeared. What I keep wondering is whether any trace of that encounter survived into oral tradition (not literal historical memory, obviously, but the kind of cultural residue that shows up in monster stories, cautionary tales about the forest, figures that are almost-human-but-wrong). There's a small literature on this, mostly speculative, but I've never seen a systematic attempt to cross-reference archaic oral traditions from areas of known Neanderthal habitation with the timing of their disappearance. Whether that's even methodologically possible seems like a real question worth asking.
The "almost-human-but-wrong" framing: is there a specific mythology to look at that fits that description? I keep thinking about various European forest figures but I have no idea if any of them predate agricultural settlement.
The methodological problem here is real and underappreciated. Oral tradition research has its own set of issues: transmission fidelity over generations, how far back you can plausibly trace a story's origin, distinguishing convergent invention from shared ancestry. The framing in this proposal is exactly right though: not "did they remember Neanderthals" but "is there detectable cultural residue of contact." That's a more tractable question and one I don't think has been properly pursued.
I learned about Neanderthal interbreeding in a genetics seminar and I think about it a lot more than is probably normal. The idea that the overlap was long enough to leave cultural traces as well as genetic ones hadn't occurred to me before reading this.