To find the surface area of a cube, we use the formula:
\[ \text{Surface Area} = 6s^2 \]
where \( s \) is the length of a side of the cube.
For the original cube with side length 8 cm:
\[ \text{Surface Area} = 6(8^2) = 6(64) = 384 \text{ cm}^2 \]
If the cube is dilated by a scale factor of \( \frac{1}{4} \), the new side length \( s' \) of the dilated cube will be:
\[ s' = 8 \times \frac{1}{4} = 2 \text{ cm} \]
Now we can calculate the surface area of the dilated cube:
\[ \text{Surface Area}' = 6(s')^2 = 6(2^2) = 6(4) = 24 \text{ cm}^2 \]
Thus, the surface area of the dilated cube is:
\[ \boxed{24} \text{ cm}^2 \]