The main theory is that the Morag Tong had redirected there worship of Mephala to Vivec (the Anticipation of Mephala) in exchange for their legal existence, and that the Dark Brotherhood was originally made up of people who objected to this and decided to break away from the Morag Tong.
"In order to exist, the Morag Tong must have appealed to the highest power in Morrowind, which at that time, the Second Era, could only have been the Tribunal of Almalexia, Sotha Sil, and Vivec. Mephala, whom the Tong worshipped with Sithis, was said to have been the Anticipation of Vivec. Is it not logical to assume that in exchange for toleration of their continued existence, the Tong would have ceased their worship of Mephala in exchange for the worship of Vivec?"...
"The Night Mother, my dear friend, is Mephala. The Dark Brotherhood of the west, unfettered by the orders of the Tribunal, continue to worship Mephala. They may not call her by her name, but the daedra of murder, sex, and secrets is their leader still. And they did not, and still do not, to this day, forgive their brethren for casting her aside."
Some people claim that since Mephala had some DB members killed, she can't be the Nightmother...but isn't setting her followers against each other for shits and giggles EXACTLY the kind of thing "The Webspinner" would love to do?