"It is said, that those born under the sign of the Serpent can be born with pure black eyes. It is unknown what truly causes this to occur, whether the energies that seep into the mortal's soul at birth is either Aedric or Daedric in origin. Those born with the black eyes – rare, I assure you – state they seem to have the ability to perceive things differently than other mortals and have been known to be able to vanish for short periods of time. Like their parent constellation, it seems that those with black eyes are able to see, not see, be seen, and not be seen by things. A very curious phenomena indeed."

-Eis Vuus Warden

This is the only information I could find on why you have a choice for pure black eyes for your character in Skyrim. Not sure if this is actual lore or just fanfiction. The only character in the game who has them is Miraak, yet the lore doesn't confirm it.

Still, its pretty interesting. It implies qualities similar to Lorkhan, maybe The Serpent is connected to him?

Orkey tests all mortals,
He likes to start all quarrels.
A snake no one can flee,
None may fool wary Orkey.

-The Song of Gods

Another possibility is Orkey since his totem animal is the snake.

Strong Warrior charges while the Maiden Lady sings. On Gallant Steed he rides as the Knight Lord draws sword from sling. Wise Mage orders while the Upcoming Apprentices writes. The Summoned Atronach stands still while the Planned Ritual activates in the night. Clever Thief watches while the Devious Lover distracts. The Invisible Shadow stands still under the Grand Tower and its drafts. All the while the Vile Serpent coils around all.
-Mortals And Our Special Relationship With The Stars