Disclaimer: Dread Ringers players need not read further!

So I'm planning on running a 3-4 session mini campaign for my player group throughout the month of October. I asked the players what they'd like to see, and at least two agreed on some Eldritch Horror/spoopyness and, well. It's October. And I love to Get Spooky™

I've done a little bit of googling, but there's not altogether a ton of good, useful, applicable advice to injecting some good Eldritch or Cosmic Horror into one's game, especially for something that's just a few sessions long.

My current ideas include tossing in some Deep Ones (there's a few decent homebrew ones on here, though I'll likely whip up something of my own) and maybe having the Final Boss be some nameless, shapeless, pandimensional aberration of some horrific description, but other than that I'm a bit short on ideas here. Thus, I've waddled here looking for suggestions!

The main thing is that it fits with, or at least originates as, a Pirate treasure hunt. I've got everything I need for my setting (worldbuilding is where I shine), just need some Spooky Stuff to stuff it full of - minding that this setting isn't normally full of spookyness, and I want to ensure it still fits in my world after the adventure is done, so it can't be some world-changing conclusion.

If you really want to go the full eldritch horror route rather than just "ooh spooky pirate ghosts", I don't think it would be too hard to  shift gears fairly quickly.

Look to Lovecraft for inspiration -- so many stories started out as exploration (which fits with a treasure hunt) before the heroes begin discovering things they wish they hadn't. Heck, the ending of the actual Call of Cthulhu story could just as easily feature pirates since it takes place in the middle of an ocean. Structuring the mini-campaign like a CoC (the TTRPG) campaign would totally work, only you sub in all the "research" elements for "D&D treasure hunt" elements where the party has to fight off monsters before finding new clues. Maybe the first bit of treasure they find, which convinces them they're on the right path, also starts afflicting them with disquieting dreams during their long rests, etc.

If your players know the tropes, they should get on board once you starting dropping them.

Van Richten's Guide to Ravenloft has some discussion on the Genres of Horror - the section genres-of-horror Cosmic Horror might be useful if you have it.
I was also reminded of the Sunless Sea computer game of the Fallen London RPG, some mission walkthroughs might provide some inspiration.

Off the top of my head I'm imagining your seafaring group is mobile with their own ship and crew... I'd tend to want to curtail that sense of freedom in a Cosmic/Eldritch Horror scenario... who knows what's happened to the crew that haven't returned after shore leave in that shadowy port of Innsmouth you just docked at... and what of the crew that have returned why that deathly fishy pallor? are they all sick is there some kind of cure?

is that an H.P. Lovecraft reference? (innsmouth, not the J.R.R tolkien quote)

make things disappear, be it in some strange fog (anything can be off about it, up to you) some tentacles, or some invaders on another ship

"that shadowy port of Innsmouth..." ooh you got me with that oh so subtle reference to The Shadow Over Innsmouth by H.P. Lovecraft :D

The best piece of advice I heard if one runs a game called GURPS: Cthulhupunk is "don't let the players know they're playing Cthulhupunk." Start out with a simple treasure hunting game, and then let the elements of the eldritch coming to be. I would even go as far as to not pitch it as a horror or Halloween themed game, and then surprise the players with Lovecraftian horror. The things that happen in Shadow over Innsmouth and The Call of Cthulhu are strange rather than monstrous, to begin with. It's that they can't be explained or that they're irrational, not that the monsters are icky as such. Think along the lines of what will make the characters paranoid, irrational and exhausted (mechanically this would be things like penalties to skill checks, being frightened or other status effects, as they can't think or act rationally), rather than taking away their hit points.

The natives they're robbing could be performing a ritual that a Religion check would only reveal is "spoken only in hushed tones within the clergy." A history check might reveal the artefact is "ancient and worn, yet made of a material and with a technique that has only recently been discovered on [other, distant location]." An arcana check might reveal of a pillar "its runes are recognisable as divination magick, but the energy it gives off is necrotic." A nature check of anything - even an object - is simply "you get an uneasy sense that this is a predator. Carnivorous." Just to give some ideas. Nothing should be certain until it's revealed itself, and even then it shouldn't be identified by name, only features. It might also be worth noting that the person I linked above also says of Cthulhu: if you stat it, they will kill it. Make the players run, perhaps surrendering their ill-gotten gains, and live to tell a tale no-one will believe. Some monsters might go down, they might even stay that way.

I hope these ideas help, and indeed those above me which are all terrific. I wish you luck in this caper. It's made me want to run such an adventure now!