Why is Brynden Rivers called the three-eyed crow?

Question

Why is Brynden Rivers called the three-eyed crow? I know that he is/was a man of the Night's Watch. That explains the crow part but where was the "three-eyed" part explained?

Answer 1

There are some mentions on the books if I remember correctly, but all I could find was in the Who is the three-eyed crow? FAQ:

First introduced in A Game of Thrones, the three-eyed crow has been a mysterious figure who opened Bran’s “third eye” to magic, and began him on a path that has led him from Winterfell to the lands beyond the Wall.

So, the third eye is the 'greenseer' eye. The other two are the normal eyes that all humans have. He is considered to be the last Greenseer, so he is the last one to have the third eye open -- that's why he is called the 'three eyed crow'

Answer 2

There are two parts to the nickname, as you appear to have worked out yourself.

Three-eyed

This is a reference to Brynden's "third eye" or his greenseer eye. I can't find a reference for this being explicitly mentioned of Brynden but we do see it mentioned a lot with Bran, for example.

"You have three. The crow gave you the third, but you will not open it." He had a slow soft way of speaking. "With two eyes you see my face. With three you could see my heart. With two you can see that oak tree there. With three you could see the acorn the oak grew from and the stump that it will one day become. With two you see no farther than your walls. With three you would gaze south to the Summer Sea and north beyond the Wall."

^{A Clash of Kings, Bran IV}

Crow

This is a reference to the fact that Brynden was sent to the Night's Watch and even rose to become Lord Commander.

The old man heard him. Though Aemon's eyes had dimmed and gone dark, there was nothing wrong with his ears. "I was not born blind," he reminded them. "When last I passed this way, I saw every rock and tree and whitecap, and watched the grey gulls flying in our wake. I was five-and-thirty and had been a maester of the chain for sixteen years. Egg wanted me to help him rule, but I knew my place was here. He sent me north aboard the Golden Dragon, and insisted that his friend Ser Duncan see me safe to Eastwatch. No recruit had arrived at the Wall with so much pomp since Nymeria sent the Watch six kings in golden fetters. Egg emptied out the dungeons too, so I would not need to say my vows alone. My honor guard, he called them. One was no less a man than Brynden Rivers. Later he was chosen lord commander."

^{A Feast for Crows, Samwell II}

Though many agreed, and were pleased to see another Blackfyre pretender removed, King Aegon felt he had no choice but to condemn the Hand, lest the word of the Iron Throne be seen as worthless. Yet after the sentence of death was pronounced, Aegon offered Bloodraven the chance to take the black and join the Night's Watch. This he did. Ser Brynden Rivers set sail for the Wall late in the year of 233 AC. (No one intercepted his ship). Two hundred men went with him, many of them archers from Bloodraven's personal guard, the Raven's Teeth. The king's brother, Maester Aemon, was also amongst them.

Bloodraven would rise to become Lord Commander of the Night's Watch in 239 AC, serving until his disappearance during a ranging beyond the Wall in 252 AC.

^{The World of Ice and Fire, The Targaryen Kings: Aegon V}

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The full name also appears to be coined by Bran and Jojen, perhaps separately I'm not sure, because of how Brynden decides to show himself to each of them; as a literal three eyed crow.

Bran looked at the crow on his shoulder, and the crow looked back. It had three eyes, and the third eye was full of a terrible knowledge. Bran looked down. There was nothing below him now but snow and cold and death, a frozen wasteland where jagged blue-white spires of ice waited to embrace him. They flew up at him like spears. He saw the bones of a thousand other dreamers impaled upon their points. He was desperately afraid.

^{A Game of Thrones, Bran III}

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Lastly, it is worth noting that this name does not appear to be in reference to the fact that Brynden only had one eye, after he lost it to Bittersteel. This does not appear to be the case because before Brynden was sent to the Wall there was a riddle that asked how many eyes does he have. With the answer being "A thousand eyes and one". The thousand being a reference to his spies and him being a "sorcerer" and the one obviously being a reference to him only having one physical eye left.

How many eyes does Lord Bloodraven have? ran the riddle Egg had heard in Oldtown. A thousand eyes, and one.

Six years ago in King's Landing, Dunk had seen him with his own two eyes, as he rode a pale horse up the Street of Steel with fifty Raven's Teeth behind him. That was before King Aerys had ascended to the Iron Throne and made him the Hand, but even so he cut a striking figure, garbed in smoke and scarlet with Dark Sister on his hip. His pallid skin and bone-white hair made him look a living corpse. Across his cheek and chin spread a wine-stain birthmark that was supposed to resemble a red raven, though Dunk only saw an odd-shaped blotch of discolored skin. He stared so hard that Bloodraven felt it. The king's sorcerer had turned to study him as he went by. He had one eye, and that one red. The other was an empty socket, the gift Bittersteel had given him upon the Redgrass Field. Yet it seemed to Dunk that both eyes had looked right through his skin, down to his very soul.

^{The Sworn Sword}