The astronomer Dr. Phil Plait, a former CSU Sonoma professor who runs the Bad Astronomy blog, is behind a new televisions how called Bad Universe, and wrote the excellent Death From the Skies, once was interviewed regarding people who believe that the Moon landing was a hoax. He stated that it was difficult to debate such people because they are not constrained by evidence, and therefore can make up a nearly infinite number of explanations or rationalizations supporting their position, and if you are to go at them one-by-one (as they try to get you to do) you will never have time to present the actual evidence. Plus, given that they can produce such a huge number of fallacious claims, one can quickly become bogged down trying to deal with the claims themselves even if you don't try to present the actual evidence.
This point seems to hold true for pretty much every form of psuedo-science and psuedo-history, whether it be moon-landing hoaxers, "Obama Birthers" (the people who believe that he's not a US citizen), vaccine denialists, young-Earth creationists, 9-11 "truthers", and the list goes on and on. Unconstrained by reality, and motivated by emotional factors and/or ideology, these folks can create an endless series of justifications for their positions, all without ever saying anything that actually stands up to scrutiny.
The same is also true for people with pseudo-scientific understandings of the human past. About once a month, I receive an email from someone or else am asked by an acquaintance about some alleged mystery or inconsistency regarding the human past. The vast majority of the time, these questions are asked honestly, the person asking them really wants to know if there is any backing to the strange claim that they've heard. But every now and again, I get one where the person is trying to convince me of the truth of a spurious claim, or on even rarer occasions they try to get me to "fess up" to the role that I have played as a member of the "establishment" in "hiding the truth." It's the members of this latter group that almost always produce the flood o' nonsense that comes so quickly that there's not enough time to fill up the sandbags of reality.
The flavors of crazy are varied. Some people are convinced of Biblical literalism and will come up with an increasingly desperate string of rationalizations for why the Bible is true in the face of overwhelming evidence to the contrary; others want to see a past that in some way reflects their view of what the political and/or social order should be - both Israelis and Palestinians jump on anything that sounds like archaeologists providing information supporting their side in the conflict; the "Ancient Astronauts" fans routinely make shit up about ancient societies (or deny things known about them) in order to support their position; I have heard those committed to naturopathy make false claims about the lifespan and healing knowledge of hunter-gatherers; people wanting to "elevate" their particular ethnic group may claim the works of another group as theirs; and the list goes on.
Regardless of the exact claims made, the same basic type of discussion plays out when you talk with such people: They will make a claim, challenge you to refute it then and there, and if you can they will offer another claim that must be refuted then and there, and so on, and so on, and so on...
While most archaeologists simply brush these people off as kooks, that doesn't seem a particularly useful approach. Some of these people are genuinely interested in the human past, and would love it if actual archaeologists were more communicative. Others buy into this stuff because only the proponents of nonsense have reached out to them, but might be open to hearing a more evidence-based take on things. However, in order to get these people to look into the actual archaeology, you are likely to have to disprove a very, very long list of false claims and honest misunderstandings. Some folks will catch on pretty quickly that much of what they have learned is false, but others will continue to push claims that you can't refute not because they make sense, but simply because you have never heard them before and therefore while you may see all of the red flags of pseudo-science, being able to respond to them in a specific way becomes difficult. The problem is rather a confounding one.
I don't know what the answer is, exactly. But I do know that the approach I usually see taken (blowing people off) reinforces the beliefs. Likewise, responding that you don't know the answer to a question but will get back to them when you have researched it is the most honest and reasonable response, but is also likely to reinforce the belief because they "stumped an expert." Even if you come back to them with a refutation later, they are likely to stick to the original false belief.
It's a weird aspect of human psychology that both encourages people to believe the absurd (especially when it coincides with their existing beliefs), and which reinforces the belief when someone knowledgeable about the subject doesn't wish to discuss it.