Step 1: Find $\sin A$ directly instead of rewriting the expression.
Rather than splitting $2\cos^2 A$ into $\cos^2 A + \cos^2 A$, let's just compute the actual value of $\sin^2 A$ first and substitute both pieces straight into the expression.
Step 2: Recall the identity needed.
\[ \sin^2 A + \cos^2 A = 1 \implies \sin^2 A = 1 - \cos^2 A \]
Step 3: Compute $\sin^2 A$ using the given value of $\cos A$.
We are told $\cos A = \frac{1}{2}$, so:
\[ \cos^2 A = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]
\[ \sin^2 A = 1 - \frac{1}{4} = \frac{3}{4} \]
Step 4: Substitute both values straight into the given expression.
\[ \sin^2 A + 2\cos^2 A = \frac{3}{4} + 2\left(\frac{1}{4}\right) \]
\[ = \frac{3}{4} + \frac{2}{4} \]
\[ = \frac{5}{4} \]
Step 5: Final answer.
The value of $\sin^2 A + 2\cos^2 A$ is $\frac{5}{4}$, which is option (B).
\[ \boxed{\dfrac{5}{4}} \]