In an LC oscillator, if values of inductance and capacitance become twice and eight times, respectively, then the resonant frequency of oscillator becomes $$x$$ times its initial resonant frequency $$\omega_0$$. The value of $$x$$ is:

The resonant frequency of an LC oscillator is given by:

$$\omega_0 = \frac{1}{\sqrt{LC}}$$

When the inductance becomes twice ($$L' = 2L$$) and capacitance becomes eight times ($$C' = 8C$$):

$$\omega' = \frac{1}{\sqrt{L'C'}} = \frac{1}{\sqrt{2L \cdot 8C}} = \frac{1}{\sqrt{16LC}} = \frac{1}{4\sqrt{LC}} = \frac{\omega_0}{4}$$

Since $$\omega' = x \cdot \omega_0$$:

$$x = \frac{1}{4}$$

Therefore, the correct answer is Option A: $$\mathbf{\frac{1}{4}}$$.