How can one construct the weakest S-box ever?
Any affine function will do. Let your Sbox be $$S(x)=Mx\oplus c$$ where $M$ is an $n\times n$ binary matrix and $c$ is an $n-$bit constant vector.

The output difference for this Sbox is, for any nonzero $a$
$$
S(x \oplus a)\oplus S(x)=(M(x\oplus a)\oplus c )\oplus Mx\oplus c= M a\oplus c
$$
which is a constant for fixed $a$ so all the output differences for that input difference take on the same value.

For nontriviality pick either $c$ nonzero or $M$ a non identity matrix. The matrix must be invertible (over $GF(2)$) for the Sbox to be invertible, i.e., a proper substitution.

AES with this type of Sbox can be trivially broken.

Edit: See the answer to this question for details.