Step 1: Electrolysis of aqueous NaCl.
During the electrolysis of aqueous NaCl, at the anode, chlorine gas (Cl$_2$) is evolved, and at the cathode, hydrogen gas (H$_2$) is produced. This is because water is reduced at the cathode.
Step 2: Analyzing the options.
(A) O$_2$(g): Oxygen is not produced at the cathode; it is produced at the anode.
(B) Cl$_2$(g): Chlorine gas is produced at the anode, not at the cathode.
(C) H$_2$O(g): Water vapor is not typically produced at the cathode during electrolysis.
(D) H$_2$(g): Hydrogen gas is produced at the cathode during electrolysis of aqueous NaCl.
Step 3: Conclusion.
The correct answer is (D) H$_2$(g) as hydrogen gas is produced at the cathode.