Solution:

Substitute $u = |x|,$ $v = |y|.$ Then $|u - 1| + |v - 1| \le 1$ is a diamond centered at $(1, 1)$ with diagonals of length $2,$ so it has area $2,$ and it sits entirely in $u, v \ge 0.$ The map $(x, y) \mapsto (|x|, |y|)$ is four-to-one over $u, v \gt 0,$ so the full region has area $4 \times 2 = 8.$ Thus, B is the correct answer.