Your evidence must be older than 50,000 BC. Considering Neanderthals first appear around 500,000 years ago, and go extinct around 30,000 years ago, I hardly think asking for evidence older than 50,000 years is unreasonable if they truly did have human-level intellect.
Why do I request this you might ask? Simple, humans can unambiguously be placed in the middle east ~50,000 years ago, and in Europe ~45,000 years ago. Because of the error bars on dates, anything younger than 50,000 years old could easily be a human artifact.
Think about it...say you are an archaeologist of the paleolithic period. You have a European necklace dated to ~40,000 years old, a time when both humans and Neanderthals lived in Europe, do you:
A. Infer that the necklace was made by early humans in Europe, seeing as evidence for such complex behaviors among Neanderthals is lacking and is abundant in modern humans after 50,000 BC.
B. Infer that Neanderthals made the necklace, evidencing a previously undocumented human behavior in Neanderthals, showing that our "prejudices" against them were wrong all along.
Obviously, the first one is the more reasonable inference, but the second one will make your paper stand out more and get you more grant money. That paper would certainly be the more exciting one, even if unfounded.
Modern human behavior explodes onto the scene in Africa, Asia, Arabia, Europe, and even Australia around 50,000 years ago. This coincides with anatomically modern humans rapidly migrating across continents. It also roughly coincides with the earliest, unambiguous anatomically modern human remains from Manot cave in Israel.
So, taking behavior that is dated to after 50,000 years ago and applying it to Neanderthals is simply unreasonable. As such, I think I am perfectly justified in rejecting any "evidence" that violates this criteria.
Before I move on to the second criteria, I want to list some lines of "evidence" that unapologetically violate this criteria (thus, I reject them as evidence, and I'm by far not the only one):
The Chatelperronian Culture
Alleged Neanderthal Cave Art From El Castillo Cave, Spain
Alleged Neanderthal Cave Art From Gorham's Cave, Gibraltar
Mammoth Bone Huts
And this is just the tip of the iceberg! Such bold claims to be made with so little evidence.
Your "evidence" cannot apply to living animals. For instance, Neanderthals used sharpened sticks as spears, Neanderthals probably "buried" their dead, Neanderthals probably used some plants as a form of "self-medication", Neanderthals probably cooperatively hunted in small groups, Neanderthals probably had some crude form of language, Neanderthals made crude stone tools, etc.
However, every single one of these behaviors is present in living animals, and is in no way evidence that Neanderthals were human beings made in the Image Of God. It's just evidence that they were higher animals, with will and emotion, as well as limited problem solving and spatial reasoning.
Chimpanzees make wooden spears to hunt Bushbabies.
Elephants and Crows bury their dead, normally in sticks and leaves, but with Elephants sometimes with actual dirt. Magpies even give flowers to their dead.
All over the animal kingdom self-medication with particular plants is present.
Lions, Wolves, Hyenas, Some Hawks, and even the extinct dinosaur Tyrannosaurus Rex hunt in groups.
Birds, Prairie Dogs, and Dolphins have a form of language.
Bonobo apes make stone tools.
Yet when similar behaviors are found on prehistoric hominids, they cite it as evidence that the creatures are human beings on the inside, and have human minds. Choosing to see it as a sort of "racism" if you consider Neanderthals to be mentally inferior.
In conclusion, if you want to have a discussion with me about the intelligence of the Neanderthals (or other hominids for that matter, as technically these criteria apply to any hominid, Neanderthals are just the ones that people most like to hold up on a pedestal), you best to keep these criteria in mind.
Bonus: Easy Response Questions:
1. This one is to show that the behavior of a hominid is actually present in living animals, demonstrating that the behavior does not indicate a human mind is present.
"So, when <insert animal that does the "smart" activity here> <insert the "smart" behavior here>, they are still animals, but when <insert hominid here> does it, they must be human beings or "near-human"?"
2. This is very similar to the last one, but is intended for behaviors that aren't identical, but are comparable. (Or even better, when the animal behavior is far more sophisticated than the alleged hominid behavior)
"So, when <insert example of very smart living animal behavior here> but when <insert hominid behavior "find" here> they must be people, not animals?"
3. This one is for exposing the faulty logic when criteria 1 is violated.
"So, <insert hominid here> were <insert "smart" activity here>", even though they were around for hundreds of thousands of years, the evidence just *happens* to show up when modern humans were already present in the vicinity of these hominids?"
These three quotes debunk approximately 90% of finds that allegedly support human behavior in hominids (and 90% is a very conservative estimate, probably higher). This should tell you something about the validity of the idea that Hominids were human beings, rather than just animals with an anatomical (and by extension, genetic) resemblance to humans.
Thanks for reading!