To calculate the surface area of a cube, we use the formula:
\[ \text{Surface Area} = 6 \times \text{side length}^2 \]
For the original cube with side length \( l = 8 \):
\[ \text{Surface Area} = 6 \times 8^2 = 6 \times 64 = 384 \]
Now, if we dilate the cube by a scale factor of \( \frac{1}{4} \), the new side length of the cube will be:
\[ \text{New Side Length} = 8 \times \frac{1}{4} = 2 \]
We can now calculate the surface area of the dilated cube:
\[ \text{New Surface Area} = 6 \times 2^2 = 6 \times 4 = 24 \]
Thus, the surface area of the dilated cube is:
\[ \boxed{24} \]