The Deep One origins of the Citadel
Peremore, the younger brother of King Urrigon Hightower, son of King Uthor of the High Tower, was (according to the official histories of the Citadel) a 'sickly boy born with a withered arm and twisted back'. While he was bedridden, he had a great thirst for knowledge, and surrounding himself with 'wise men, teachers, priests, healers, and singers, as well as wizards, alchemists and sorcerers'. When Peremore died, his brother the king donated a large tract of land to "Peremore's pets", to continue their quest for knowledge about the world.
However, given that the official history of the Citadel (and of Oldtown and the Hightowers) is very shady, I question this, and propose an alternate theory: Peremore was a Deep One Hybrid, born of a human mother who was impregnated by a Deep One. He, like many of these hybrids, was born a monstrosity, but he inherited the scholarly nature of the Deep Ones.
What evidence is there of the Deep One origin of Peremore?
Let's examin the origin of his name:
"Pere-", as in peregrine falcon, derives from the Latin per, which means through. When combined with grine or its original grinus (which is from the Latin ager, meaning field), the word means they came through that field to get here. A peregrine falcon is the wandering falcon; more broadly, a peregrine object is a wandering, or foriegn, object.
More, or mor, is a word of Celtic origin meaning sea and is related to the French mer and the Middle English mere-, as in merman or mermaid. The Celtic Sea (the area off the south coast of Ireland) is known as Mor Celtaidd in Welsh, An Mhuir Cheilteach in Irish, Y Mor Celtaidd in Cornish. Mor is also a Celtic goddess of the sun and sea.
With this understanding, then, we see that the name of the founder of the Citadel, Peremore, can be translated as through the sea or from the sea.
Deep Ones, within the Cthulhu Mythos, are a separate species that live in the oceans of Earth. While there are some hints of an extra-terrestrial origin for them, its most likely that they evolved on Earth, according to the Encyclopedia Cthulhiana.
The lives of the Deep Ones are immortal. Unless they are murdered, they will live forever. Many Deep Ones use their extended lives to become reknowned scientists or priests.
Following in these footsteps, Peremore became a scholar, studying the humans and beasts of Westeros. He surrounded himself with other scholars, the famous "Peremore's pets" to whom King Urrigon bequeathed the lands of the Citadel.
The tales of the Squishers in A Feast For Crows describes in detail the Deep Ones of Terros (the land of Ice and Fire, also known as Earth, as GRRM recently made clear). The immortality of the Deep Ones is a legend along the east coast of Westeros:
"Dick: They'd eat you, boy. They'd eat you raw. Podrick: If they try, I'll kill them. Dick: You try that. You just try. Squishers don't die easy." - A Feast For Crows, George RR Martin
While the First Men supposedly killed all of the Squishers (likely using runic armor to protect themselves), Dick Crabb insists that they still live and eat human men while stealing human women, with which they mate.
Hybrids and monstrosities
"The Deep Ones are a race of intelligent ocean-dwelling creatures, approximately human-shaped but with a fishy, froggy appearance. They regularly mate with humans along the coast, creating societies of hybrids."
The human cults that worship the Deep Ones focus on the matings between Deep One fathers and human mothers. Most successful hybrids grow up looking human, though with a particular fishy look to their face. Upon becoming adults, most become embrace their Deep One nature and join their fathers in the underground cities of the Deep Ones.
In H.P. Lovecraft's The Shadow Over Innsmouth, which is the debut of the Deep Ones within the Cthulhu Mythos, he tells of a town in Massachusetts that has made a contract with the Deep Ones. In exchange for plentiful fisheries and secret gold artifacts, the townspeople interbreed with the Deep Ones.
The children of these matings initially look like normal humans, but start to look like their Deep One ancestors as they age. This is known as the 'Innsmouth Look'.
Here is a list of the basic features of these hybrids:
Shrunken/nonexistent ears
Glassy, bulging, unblinking eyes
Folds on the neck which eventually become functional gills
Once these hybrids become too alien in appearance, they hide themselves in secluded, abandoned buildings, until such time as they complete their metamorphasis and desire to join the Deep One city of Y'ha-nthlei.
Many of the children born of the mating between Deep Ones and humans are born as monstrosities, according to the Encyclopedia Cthulhiana. Peremore fits this mold: his withered arm and twisted back indicate that a mating was not fully successful, and he died as a child, not achieving the immortality of the Deep Ones which comes to hybrids in middle age.
Conclusions and speculations
If (as I suspect) the histories of the Citadel are not telling the full story of the founding of their order, then there is plenty of room to speculate about Peremore, and indeed, the whole of the Hightowers (who I have addressed in other essays). Given the origin of his name and his Deep One-like desire for knowledge, the story of Peremore presents a possibile Deep One origin for the Citadel.
There is also the possibility that Peremore did not die, but instead either lives on at the Citadel in seclusion or joined the Deep Ones in an underwater city, as the hybrids would often become secluded until they became fully realized as Deep Ones.
Deep Ones are everywhere in Terros
The Deep Ones are a pervasive part of A Song of Ice and Fire. The squishers of Crackclaw, the webbed hands of House Borell, the peoples of Toad Isle and their fishy appearance, the terrified green skinned people of the Thousand Isles, the riddles of Patchface, and most importantly, the speculated origins of the Ironmen, all demonstrate that the Deep Ones surround, and interact with, the humans of Westeros and Essos.
Controlling the knowledge of Westeros through their order, the Deep Ones are able to keep tabs on the activities of the humans, and played a great part in the downfall of the Dragons. The Deep Ones control the communications on Westeros, and plot against the Targaryens in ways that don't make much sense... unless they have a deeper purpose.
While I'm not totally sure of their purpose yet, I believe that the plot detailed in The Rogue Prince and The Princess and The Queen indicate a relationship between Harrenhal (founded by the Deep One-derived Ironborn) and the Citadel. The Hightowers and the Whents (who were still in control of Harrenhal in these early stories) conspired with the Citadel to bring about the Dance of the Dragons, which destroyed the dragons of the Targaryens. This lays the blueprint for a similar conspiracy to occur within the current storyline.
The Deep Ones played a greater part in the wars and battles of the Dawn Age than is let on, and that they will play a part in the prophesizied second Long Night cataclysm. It is more than just knowledge that they are after; instead, they desire to conquer humanity.
After the last Long Night, the humans and Children of the Forest banded together to banish the Deep Ones to their oceans, but the end of the banishment is nigh, and they will return.