Pardon my attempt at being different here but I often feel that keeping this mysterious nature around concepts that most scientists with Phd know about is a little ironical, the true non-proliferation mechanism is the fact that getting the materials needed for a bomb is very complicated both logistically and arranging them technically not the basic aspects of nuclear engineering which I think are known to many even outside the field of bomb production.
@cmb Well I don't know how far you understand the basic physics here but the secondary which is the fusion part of a thermonuclear weapon is located physically separate from the primary in most designs, so as the primary goes off the first thing it does it releases lots of Xrays mostly also gamma. Em radiation is faster than physical shock waves so now you have this released radiation which you need to use to rapidly heat up the secondary before the physical shock has turned the bomb into dust, all of this happens within the so called radiation waveguide which is the bomb casing inner side.
I think it's called waveguide because of the similarity to RF waveguide phenomena only here one guides a photon flux, even though it's hard to "guide" high energy photons, many due get scattered and reflected against the high Z (atomic number) heavy metal bomb casing walls.
As this happens this photon flux also impacts the secondary. The outer portion of the secondary is also made from a high Z material like U 238, often called a tamper. Something interesting happens here. As the photons hit the heavy tamper material they heat it up and eject it's atoms, aka ionize the tamper, as this is happening the tamper is both heating up and being ablated away (also called tamper ablation) this photon flux pressure as well as matter ejection recoil causes the tamper sphere to implode rapidly.
This compresses the inner fusion fuel to fusion conditions.
But to answer your core questions. Think about it this way. Both the primary and secondary of the bomb are within the bomb casing and they are rather heavy but they need to be held up so that they don't fall down and roll around like candies within a box inside the bomb. In order for the implosion to go as planned the secondary needs to be positioned correctly so that the photon flux can ablate the tamper evenly, if you screw the implosion you don't get the full yield of the bomb and many other problems arise. You need a material that is as transparent to the photon flux as possible so that as many photons from the primary reach the secondary as possible.Also you need the material to have considerable strength and not break apart during handling and storage of the bomb.
This I think is why it's called Fogbank.
It could be some sort of a specific aerogel , but before aerogel or any material x came about I think they used some plastics like
@anorlunda said polystyrene etc.
PS. I don't think Soviet security forces created the idiotic red mercury joke, I think it was just a brilliant scam used by smart Russians to scam gullible and stupid westerners into buying worthless cold war era military surplus stuff like old outdated radios, by saying that the vacuum tubes inside those radios contain the "supersecret" voodoo God like substance :D
It's just an urban legend that some used to get rich quickly.
Believe it or not I once sold an old USSR radio which was quite popular around here, many potential customers on Ebay asked me can I open up the box and photograph the tubes, I asked why? They told me they are looking for a red substance in them...
:D
This right here ladies and gentleman are how hoaxes and conspiracies are born, all I had to do is say to them that indeed all the tubes are just overflowing with red liquid.
One guy was willing to pay me twice the asking price, I'm not joking this is true, I was amazed myself at how uneducated and lacking in critical thinking some folks can be.