Hmm context of the death threat might be a releavnt detail, If I knew the person was joking or if I had any relationship where I actually knew stuff about them, I probably wouldn't.
But on the other hand if it was serious, or if the person was a complete stranger, chances are most likely I would try to get some money out of it.
I'd only sue if you fealt that your life is in actual danger from the said death threat, but if that fails, you might just piss them off enough to carry out the threat