To find $f^{-1}(0)$, we need to find the value of $x$ that makes $f(x) = 0$.

Step 1: Set $f(x) = 0$:
\begin{align*}
2 \log_2 (2x - 1) &= 0 \\
\log_2 (2x - 1) &= 0
\end{align*}

Step 2: Solve for $x$:
\begin{align*}
2x - 1 &= 2^0 \\
2x - 1 &= 1 \\
2x &= 2 \\
x &= 1
\end{align*}

Step 3: Verify the answer:
\begin{align*}
f(1) &= 2 \log_2 (2(1) - 1) \\
&= 2 \log_2 (1) \\
&= 2 \cdot 0 \\
&= 0
\end{align*}

Thus, $f^{-1}(0) = \boxed{1}$.