To find the unit rate in minutes per meter, we first determine how many meters the aquarium is filled per the time it takes to fill it.
The aquarium is filled to a depth of \( \frac{5}{8} \) meters in 5 minutes.
Now, we can calculate the unit rate by dividing the total time by the depth filled:
\[ \text{Unit Rate} = \frac{\text{Time}}{\text{Depth}} = \frac{5 \text{ minutes}}{\frac{5}{8} \text{ meters}} \]
To divide by a fraction, we multiply by its reciprocal:
\[ \text{Unit Rate} = 5 \text{ minutes} \times \frac{8}{5} \text{ meters}^{-1} \]
The 5s in the numerator and denominator cancel out:
\[ \text{Unit Rate} = 8 \text{ minutes per meter} \]
Thus, the unit rate of filling the aquarium is \(\boxed{8}\) minutes per meter.