Step 1: Species present in very dilute NaCl solution.
The solution contains $Na^+$, $Cl^-$, and $H_2O$. In very dilute solution, $Cl^-$ concentration is negligibly low.
Step 2: Cathode reaction.
Reduction of $Na^+$ requires very high energy. Water is preferentially reduced: $2H_2O + 2e^- \rightarrow H_2 + 2OH^-$. So $H_2$ is evolved at the cathode.
Step 3: Anode reaction.
At low $Cl^-$ concentration, water is preferentially oxidized: $2H_2O \rightarrow O_2 + 4H^+ + 4e^-$. So $O_2$ evolves at the anode, not $Cl_2$.
\[ \boxed{H_2 \text{ gas is evolved at cathode}} \]