"In order to determine who the first born is, we should look to the source material that Dark Souls draws the most from, Shinto mythology.
The most prominent first born of Shinto myth is that of Hiruko, the son of Izanami and Izanagi. Izanagi and Izanami are known to have fathered many beings, two of the most note being Amaterasu, Goddess of Sunlight, as well as Tsukuyomi, the moon god. I hope I need not point you to the correct parallels drawn from here.
Now, the thing that must be made of note was that Hiruko was born deformed, he had no bones within his body, if the first born is not incredibly obvious by now, you must be mentally impaired. But for the sake of a complete explanation, I will continue.
Izanami later died in child birth, the one she gave birth to was Kagu-Tsuchi, the god of fire.
There is only one female in the story who was close to Gwyn, gave birth to a deformed child, and died creating a being of fire.
Therefore, Ceaseless Discharge is the first-born of Gwyn, and the Witch of Izalith, his former wife."
Not sure who came up with this, or whether its been posted or not, but its a plausible theory, and i figured you'd all like a crack at it
i know this is obvious, but the "sun goddess" and "moon god" are gwynevere and gwyndolin, for those who didn't catch that