Step 1: Understanding the Concept:
In electrolysis of an aqueous salt, water competes with the ions. At the cathode, the species that is reduced most easily (higher reduction potential) wins.
Step 2: Possible Cathode Reactions:
\(\text{Na}^+ + e^- \rightarrow \text{Na}\) has \(E^{\circ} = -2.71\) V.
\(2\text{H}_2\text{O} + 2e^- \rightarrow \text{H}_2 + 2\text{OH}^-\) has \(E^{\circ} = -0.83\) V.
The water reaction has the less negative potential, so it occurs.
Step 3: Product at the Cathode:
Hydrogen gas bubbles out of the cathode, and the solution near it becomes basic because of \(\text{OH}^-\) (this is the basis of the chlor-alkali process).
Step 4: Check the Other Options:
\(\text{Cl}_2\) forms at the anode, not the cathode. \(\text{O}_2\) is an anode product of water oxidation, and it is not formed at the cathode. Sodium metal is not deposited because \(\text{Na}^+\) is harder to reduce than water. So (D) is correct.
Final Answer:
Hydrogen gas is evolved at the cathode, option (D).
\[ \boxed{\text{(D) } \text{H}_{2(g)}} \]