It’s helpful to distinguish two types of mysterian: (1) Those who both yearn for, and depend upon, mysteries. (2) Those who discover what they deem to be “mysteries” through philosophical analysis and looking at the available science. Mysterians of type (1) don’t want science to even consider that which they believe to be (essentially) mysterious. Chomsky, McGinn, Fodor, Pinker, etc. are mysterians of type (2).
The term “mysterianism” was coined by the American philosopher Owen Flanagan. He first used the phrase “the new mysterians” in his 1991 book The Science of the Mind.
Flanagan also calls this position “anti-constructive naturalism”. The New Mysterian Colin McGinn, on the other hand, calls it “transcendental naturalism”.
Readers must now note the strong distinction which can be made between not explaining (or solving) mysteries today, and not being able to explain (or solve) the same mysteries ever.
Many (i.e., not all) Mysterians believe that the mysteries will never be explained.
The Mysteries and Cognitive Closure

Today, the superlative mystery is the (to use pretentious Platonic capitals) Hard Problem of Consciousness. This is what most mysterians have concentrated upon in recent decades. However, the primary mysterian — Colin McGinn — does class the problems of the identity of the self, the foundations of meaning, and the mind-body problem as mysteries too.
As another example, take McGinn’s words on free will:
“Free will, for instance, looks upon early inspection to be impossible, so we try to find some conception of it that permits its existence, but this conception always turns out to be dubiously reductive and distorting, leaving us with the unpalatable options of magic, elimination or quietism. [ ] so we hop unhappily from one unsatisfactory option to the next; or dig our heels (squintingly) into a position that seems the least intellectually unconscionable of the bunch [ ].”
If readers were to accept McGinn’s account above, then that would clearly show us why free will — and many other “mysteries”— have been the (to use a common phrase) “perennial problems of philosophy”.
So how do the mysterians know that human beings don’t have the cognitive means to solve (or explain) the various mysteries (or “hard problems”) at a scientific level?
At first glance, this seems very odd.
Mysterians place much emphasis on the limits of scientific knowledge. (Perhaps also on knowledge generally.) So it’s odd that they (seem to?) know that (to repeat) human beings haven’t the cognitive means to solve (or explain) the various mysteries (or “hard problems”) at a scientific level.
Of course, some mysterians have (kinda) tackled this. They claim that their (seemingly?) a priori claim isn’t an assumption or presupposition. Instead, it’s a philosophical conclusion brought about by much philosophical analysis and thought.
Mysteries Forever?

It may be helpful to distinguish two types of mysterian here:
(1) Those mysterians who both yearn for, and depend upon, mysteries.
(2) Those who simply discover what they deem to be “mysteries” through philosophical analysis and looking at the available science.
I would class the philosopher Colin McGinn, for example, as an example of (2). He’s claimed that consciousness is “a mystery that human intelligence will never unravel”. Yet he’s also keen to tell us that his position, as well as his specific claims, are naturalistic. Moreover, Noam Chomsky, Jerry Fodor, Steven Pinker, etc. have been classed as a mysterians too. However, as mysterians, they tend to focus on a single mystery: the Hard Problem of Consciousness (HBC).
Why does McGinn use the word “naturalistic” about his own position?
McGinn acknowledges the possibility that solutions to the mysteries may be knowable to an intelligent alien from another galaxy. Thus, it’s human brains that suffer from what he calls “cognitive closure”.
Let’s go into detail in why McGinn deems his position to also be naturalistic.
Colin McGinn’s P

McGinn talks in terms of a property P which would (or could) explain the link between the brain and consciousness. He explicitly states that he resolutely shuns what he calls “the supernatural”. Thus, P must be “natural”. He goes on to say that
“it must be in virtue of some natural property of the brain that organisms are conscious”.
McGinn then concludes:
“There just has to be some explanation for how brains subserves minds.”
Why must P be natural?
Is this a case of McGinn having faith in naturalism?
Alternatively, perhaps McGinn’s theory of P — or his general mysterianism — isn’t naturalistic in the first place.
In detail.
McGinn’s (or our) P is a real (or actual) property. It may even be concrete (e.g., part of the brain). Thus, McGinn doesn’t believe that he’s (to use his own term) “irrealist” about P. However, McGinn does cite the possibility that P is noumenal. As with Kant, noumena are permanently unknowable — by definition. (Kant also believed — living before Darwin — that the structures of the mind-brain would remain static for… well, the rest of time.)
It’s the very nature of the mind-brain link which renders P permanently closed off to us. Yet despite that possibility, P could still be part — at least according to McGinn — of a respectable naturalistic theory.
Thus, McGinn seems to argue that we shouldn’t be… well, mysterious about P. That said, one can’t help but see P as being mysterious if it’s also believed to be permanently closed off to us.
Scientism

Owen Flanagan told us (way back in 1992/97) that the
“new mysterianism is a postmodern position designed to drive a railroad spike through the heart of scientism”.
This passage brings forth a rather predictable binary opposition: mysterianism vs scientism.
Yet one can be against mysterianism-in-the-round without also being “scientistic”. That said, virtually no one admits to being a scientistic philosopher or individual. On the other hand, many religious (or spiritual) mysterians class almost everyone under the sun (outside their own camp) as being “scientistic” (as well as “materialist”).
Alternatively, if one is against mysteries, then, according to some religious/spiritual mysterians, you’re scientistic by definition.
So is there is some kind of conflation here of being critical of mystery-mongering and scientism?
In strong terms, this is how the critics of scientism see things:
“[Scientism is] an exaggerated trust in the efficacy of the methods of natural science applied to all areas of investigation (as in philosophy, the social sciences, and the humanities). [] [S]ome scholars, as well as political and religious leaders, have also adopted it as a pejorative term [].”
Scientism can be seen as a position (or stance) on the huge importance of science to philosophy, and — often more broadly — to just about everything else. Thus, this position needn’t be taken to be a philosophy which advances independent arguments (or positions) on philosophical subjects. Indeed, if it did do such things, then it would be like a position within philosophy (just like idealism or physicalism)…
Naturalism, on the other hand, is a position within philosophy.
Scientism, therefore, can be deemed to be over and above naturalism.
It can be argued, then, that one can be a philosophical naturalist without displaying strong scientistic leanings. Such a philosopher can be broadly naturalistic in his philosophy, yet not strongly scientistic. Indeed, he/she may know little science, and therefore not feel the need to consult the findings of science every time he/she tackles a philosophical issue.
A scientistic philosopher, on the other hand, would (or may) argue that the sciences must always be consulted when it comes to virtually every issue.
Thus, scientism surely requires an explicit and self-conscious commitment to the sciences and their role in literally all subjects.
There is a third type of mysterian. Such mysterians are classed as “pseudo-mysterians”.
Pseudo-Mysterians

They hold that consciousness cannot be grasped by human beings at present. However, consciousness may be grasped when new advances in science and technology come about. The “naturalistic dualist” philosopher David Chalmers states that he holds this position, and so too does Rebecca Goldstein. [See here.] (Whether or not these two philosophers are mysterians is another matter.)
It’s hard to see how this position is mysterian at all. Surely all physicalists — and even “scientistic” people — could also accept it. After all, which scientist or philosopher claims the following? —
Consciousness is, at present, completely and fully within the grasp of human understanding.
Consciousness may not be completely within our grasp today, but it may be in the future. Yet this is what pseudo-mysterians also believe! That is, they believe that consciousness may be comprehensible as a result of the future advances of science and technology…
But hold on here.
Some anti-mysterians may hold that the Hard Problem of Consciousness (HPC) is a pseudo-problem. That is, they may believe that it has a design that makes it (scientifically) unsolvable. However, that’s not because consciousness actually is a mystery. It’s because the (articulation of) the HPC brings forth the Hard Mystery!
Again, the HPC may be designed not to be solved by science… and possibly not by philosophy either. Thus, any future advances of science and technology won’t have the slightest impact on the HPC!
Nothing will.
Despite all the above, and as already stated, Colin McGinn — among others — claims that his own brand of mysterianism is entirely naturalistic.
Of course, that depends on definitions.
So let’s now move to those certain other strands of mysterianism which can easily be tied to supernaturalism and religion.
Religious/Spiritual Mysterians

So this is more on classification (1) above: Those mysterians who both yearn for, and depend upon, mysteries.
Some mysterians — and other people — yearn for mystery. They don’t want science to even consider that which they believe to be (essentially) mysterious.
Because of this mindset, once one mystery has been dealt with, then the mysterians clammer to invent another mystery. (At present, that mystery is consciousness, formerly it was life, the mind, etc.)
Mystery-in-itself is important to such people.
Naturalists can happily accept mysteries if they’re defined in this way:
Mysteries = Aspects of the Universe, nature and life that science hasn’t successfully — or fully — explained at this present stage in history.
However, they don’t accept this:
Mysteries must be protected from science.
Let’s trace a little of the history of the religious/spiritual mysterians.
According to Owen Flanagan:
“The ‘old mysterians’ were dualists who thought that consciousness cannot be understood scientifically because it operates according to nonnatural principles and possesses nonnatural properties.”
Here Flanagan ties the “old mysterians” to the dualists. The dualists, in turn, believed in “nonnatural principles” and “nonnatural properties”. [See note.]

Are nonnatural properties also supernatural?
In many cases, yes.
This depends, however, on how we define the prefix “super”. After all, in this context at least, “super” means “above”, “over” or “beyond” the natural, and beyond science itself. More relevantly, the full term — “supernatural” — has strong and ancient connections to religion and religious belief.
Note:
(1) It may seem odd that mysterianism is sometimes seen as “a form of nonreductive physicalism”. That’s mainly because idealists and spiritual philosophers deem physicalism to always be “scientistic”, whichever form it takes.
(*) [See my ‘Old and New Mysterians: Huxley, Tyndall, Chomsky, McGinn, Pinker’.]