In the books, the Great Other is mentioned as a enemy to R'hllor, the god of light, by Melisandre. Followers of R'hllor believe that there are only two gods, R'hllor and the Great Other, who wage an eternal war over the fate of the world. All forces of darkness, cold, and death are believed to be only servants to the Great Other. Melisandre refers to the Others as the "cold children" of the Great Other. The Great Other has never been mentioned by any Westerosi although all know of the Others from legends. According to Melisandre, sleep is a little death, dreams are whisperings of the Other who would drag everyone into his eternal night.
But we see Bloodraven and Bran:There he sat, listening to the hoarse whispers of his teacher. 'Never fear the darkness, Bran.' The lord's words were accompanied by a faint rustling of wood and leaf, a slight twisting of his head. "The strongest trees are rooted in the dark places of the earth. Darkness will be your cloak, your shield, your mother's milk. Darkness will make you strong."
This ties the Last Greenseer, Brynden Rivers, the Old Gods, and Bran Stark to the theme of darkness that draws The Great Other into opposition with R'hllor. And Melisandre sees a vision in her flames: At the Wall, when gazing into the fires, Melisandre sees a wooden face, corpse white, a thousand red eyes, and a boy with a wolf's head beside him. She thinks to herself that they must be the Great Other's champions, as King Stannis is hers.
A wooden face, therefore a face of a weirwood, a symbol of the old gods, and the three-eyed raven, which are both closely related to Bran. A corspe white face, which Bran has in the books,or could be referring to the white wood of the weirwoods. A thousand eyes, which would be the eyes of the weirwoods, and in the Dunk and egg books, you will find:
'How many eyes does lord Bloodraven have? A thousand, and one.’
A boy with a wolf's head, which would be Bran,because his sigil is the direwolf. This could mean that either Bran, or Bloodraven could be the great other, if there was such a thing.
Also, Melisandre thinks that dreams are whisperings of the great other, so that it might be true.
Another theory is that the great other, as it is called, is not evil. Perhaps, Bran or Bloodraven will be the ones to kill Melisandre, because the red god in the books can easily be portrayed as evil, and as Jon describes it, "the hungry red flame."
It can be possible that the great other has nothing to do with the other, or the night king, because the great other is not heard from in westeros, where the others live,and it can just be a name, and for all we know, the great other has done nothing evil...yet.
I'm looking forward to Stannis and Melisandre vs Bloodraven and Bran.