I'm not sure how they came up with it entirely, however it seems pretty simple.
A "Resident", is one who resides in a particular place permanently or for an extended period of time. And "Evil" is defined as: morally reprehensible, sinful , wicked, causing discomfort or repulsion, causing harm. You put them together and you get a meaning that matches the events/plot of Resident Evil itself. Get it? The monsters are evil, and the places where we find our protagonists are also the places where this evil is staying. Thus, Resident Evil.