Sodium cannot be extracted by the electrolysis of brine solution because

Brine solution is NaCl+H₂O

Electrolysis: 

$$\begin{gathered} \mathrm{NaCl}\to {\mathrm{Na}}^{+}+{\mathrm{Cl}}^{-} \\ {\mathrm{H}}_2\mathrm{O}\to {\mathrm{H}}^{+}+{\mathrm{OH}}^{-} \end{gathered}$$

Reduction potential order of cations : H⁺ > Na⁺
Oxidation potential order of anion : ${\mathrm{Cl}}^{-}>{\mathrm{OH}}^{-}$ 

→ Cation which is having high reduction potential is depositing or liberating as gas  at cathode
At Cathode [Reduction] : H⁺ + e^- → 1/2H₂
At anode [Oxidation] : ${\mathrm{Cl}}^{-}\to \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\mathrm{Cl}}_2+{\mathrm{e}}^{-}$

                                 ${\mathrm{Na}}^{+}+{\mathrm{OH}}^{-}\to \mathrm{NaOH}$

→ Anion which is having high oxidation potential is depositing (or) liberating as gas  at anode
In the extraction of Na metal from brine solution H⁺ is more easily under reduction at cathode than Na⁺ ion (or) in another words Na is being move electropositive element than hydrogen, therefore H₂ is liberated at cathode and not sodium.