Welcome to Gaia! ::

The Betrayer of Hope, over the course of the series, has gone through several names: Ishamael, Ba'alzamon, Moridin. There are others before these, and even between them, to the point where it is said not even he remembers his original name, Elam Morin Tedronai.

Those first three, however, are the most significant to me, because there are some very specific and interesting events that around the same time.

The name Ishamael, meaning the Betrayer of Hope in the Old Tongue, was given to Elam during the War of the Power, when he turned away from the Light, becoming one of the Chosen.

When the Bore was sealed by the Hundred Companions, Ishamael was only partly contained, and thus managed to return to the world on a cyclical basis for the next several thousand years. Sometime during, or maybe before, the Trolloc Wars, however, he stopped going by that name. He took up the name Ba'alzamon, meaning the Heart of the Dark in the Trolloc tongue.

After Rand kills Ba'alzamon (it was book 3 that he actually died, I believe, in the Stone of Tear), the world has a brief respite from Elam's machinations. Until Moridin arrives.
A new face, a new name, but the same man.

Other clues are given in the dealings with Aran'gar and Osan'gar. When they were given new bodies by the Great Lord, they were also given new names, and they accepted this, answering to them, and even giving them, in response to the other Chosen.

I believe that when the Great Lord gives a person a new body, he must also give them a new name, for the person that existed in the old body no longer truly exists. I also believe that the person the name is given to accepts the name out of necessity. It BECOMES their name. It becomes a part of them, and of their oath to the Shadow.

I don't know how many times Elam has been reborn, but almost certainly more than once, and the fact that he no longer even remembers his original name draws me to believe that it is no longer a part of him. It is no longer HIS.

Elam Morin Tedronai is dead. Moridin lives.

I also agree. except on one part: i think the reason that they (osan'gar and Aran'gar) were renamed was because they failed the Dark One, thus the reason for their new names. if you remember, their new names, are the names of the right hand and left hand dagger in a short lived dueling game in the AoL, (age of Legends) the reason the game was short lived, was because eventually, both contestants would die. i think thats implying somthing.