He does actually have a 'human' name - it's Rhavin Ironheart.
But long ago, he faced a difficult choice: whether to kill his own son, who had become an oppressive tyrant, or let him live. Rhavin's 'human' priorty would have been to let his son live, but his 'immortal' responsibility was to kill him and protect other people from his sons' cruelty.
He chose to kill his son, and thus felt that he cut ties with his 'human' self. So he took on a different name - Soul Reaver - as a symbolic way of breaking those ties.
He doesn't usually tell people his human name, so very few people know that he was once called Rhavin Ironheart.