Step 1: Picture the two points as two corners of a right triangle instead of jumping into the distance formula.
The points are $(4, p)$ and $(1, 0)$. Moving from $(1, 0)$ to $(4, p)$, we move 3 units horizontally (since $4 - 1 = 3$) and $p$ units vertically. These two movements are perpendicular to each other, so they form the two legs of a right angled triangle, with the straight line distance between the points as the hypotenuse.
Step 2: Apply the Pythagorean theorem directly to this triangle.
For a right triangle, (horizontal leg)$^2$ + (vertical leg)$^2$ = (hypotenuse)$^2$. Here the hypotenuse is the given distance, 5.
\[ 3^2 + p^2 = 5^2 \]
Step 3: Solve this equation for $p$.
\[ 9 + p^2 = 25 \]
\[ p^2 = 25 - 9 = 16 \]
\[ p = \pm 4 \]
Step 4: Sanity check using the well known 3-4-5 triangle.
A right triangle with legs 3 and 4 always has hypotenuse 5, which is exactly the classic 3-4-5 Pythagorean triple. Since our horizontal leg is 3 and the hypotenuse is 5, the vertical leg has to be 4, confirming $p = \pm 4$.
Final Answer:
The value of $p$ is $\pm 4$, which corresponds to option (A).
\[ \boxed{p = \pm 4} \]