Analysis

This problem involves finding the domain of a composite function. We can determine the domain by solving the inequality $1 < 2x - 1 < 3$.

Step-by-Step Solution

Step 1: Given that the domain of $f(x)$ is $(1, 3)$, we want to find the domain of $f(2x - 1)$.

Step 2: For $f(2x - 1)$ to have meaning, we must ensure that the input ($2x - 1$) is within the domain of $f(x)$. In other words, we need to satisfy the inequality $1 < 2x - 1 < 3$.

Step 3: Solve the inequality:

$1 < 2x - 1 < 3$

Add 1 to all parts of the inequality:

$2 < 2x < 4$

Divide all parts of the inequality by 2:

$1 < x < 2$

Step 4: Thus, the domain of $f(2x - 1)$ is $(1, 2)$.

Therefore, the correct answer is $\boxed{(1, 2)}$.