The first thing I'd suggest is this: The soul is the arena of moral drama in our lives.
Every day and every moment we're in a moral drama, playing a high stakes game. And what is won or lost in this game is our very soul. When we say "soul" we name this fight, this struggle. There's a quote from William James that describes what I'm talking about:
If this life is not a real fight, in which something is eternally gained for the universe by success, it is no better than a game of private theatricals from which one may withdraw at will. But it feels like a real fight.
Where in us does this moral drama take place? You could say this moral drama is taking place in the brain. But that seems inadequate. Because the stakes of this game, we would say, is "for our very soul." Something sacred and integral to my identity is in play and at risk. We fight to save our souls.
The soul is also the place where we experience moral damage. When we act in ways that violate our deeply held and most cherished values, or when we witness things that morally traumatize us, we feel that the soul is wounded, seared, or scarred. Again, the location of this damage isn't biological. The brain isn't hurt by moral wounds. It is the soul that is hurt.
So, this is some of what we mean when we say "soul." The soul names the arena of moral drama in our lives, and the soul is where we experience moral damage. We are in a real fight. We are trying to save our souls.