To determine the value of the shunt resistor required to convert the galvanometer into an ammeter capable of measuring a current of $$8 \mathrm{~A}$$, we need to use the concept of shunting where the additional resistor (shunt) is placed in parallel with the galvanometer.

The formula for calculating the value of the shunt resistor $ R_s $ is given by:

$$ R_s = \frac{R_g \cdot I_g}{I - I_g} $$

Where:

$ R_g $ = resistance of the galvanometer = $$10 \Omega$$

$ I_g $ = full-scale deflection current of the galvanometer = $$3 \mathrm{~mA}$$ = $$0.003 \mathrm{~A}$$

$ I $ = total current to be measured = $$8 \mathrm{~A}$$

Substituting these values into the formula:

$$ R_s = \frac{10 \Omega \cdot 0.003 \mathrm{~A}}{8 \mathrm{~A} - 0.003 \mathrm{~A}} $$

Perform the calculations:

$$ R_s = \frac{0.03 \Omega \cdot \mathrm{~A}}{7.997 \mathrm{~A}} $$

$$ R_s \approx 3.75 \times 10^{-3} \Omega $$

Thus, the value of the shunt resistor should be:

Option A: $$3.75 \times 10^{-3} \Omega$$