It's more or less just up to the author, editor, marketing team, or whoever else is responsible for a title. If they like the sound of it and think it works best, that's what they'll go with.

Why A Salesman? I dont know what Miller thought, but a makes him sound like he's just one of many. It makes him sound anonymous and unimportant. He's just a salesman, he's just passing through.

What about Death of Salesman? That sounds like it must be about someone whose name or nickname is Salesman. You don't typically drop articles in the middle of titles like that.

You can drop them from the beginning, though. The Death of... and Death of... work equally well, though someone choosing a title might feel strongly about one or the other. It's hard to know why they think that way, and many writers in their position would disagree and pick the other.

When the and a(n) begin a title, they are typically dropped if sorting alphabetically. The Night of the Hunter is found under N, not T. Otherwise A and T would be overloaded with titles, and--worse--everyone would have to remember which titles began with an article and which didn't. In reality many people aren't too bothered by it, and if you hear a discussion of Arthur Miller you're likely to hear someone call the play The Death of a Salesman, while a fan of old Hollywood might refer to Night of the Hunter.

The is far more common at the beginning of a title than a(n), so it's unusual for someone to misremember a title as beginning A...