Home-made way to detect lye in solution
The easiest way to measure the pH of a solution is to use an acid-base indicator. Limited to household supplies only, I would suggest a solution made from red cabbage, which should turn blue-greenish in the alkaline region.

The ions you actually get from your reaction is a higher concentration of $\ce{Na+}$ and $\ce{OH-}$, due to the fact that you are removing $\ce{H2}$ and $\ce{Cl2}$ (from $\ce{H+}$ and $\ce{Cl-}$ ions respectively) via the performed electrolysis. Under no circumstances would you generate a $\ce{Na2}$ species, you would rather write it as $\ce{2Na}$. Also, in aqueous solutions the $\ce{NaOH}$ most certainly dissociates. So speaking of the formation of solid $\ce{NaOH}$ from $\ce{Na}$ in water is quite a stretch.

Bottom line: Since the salt is already dissolved when you start the electrolysis, all you get is a change in concentration of the ions involved. The thereby induced change in pH can be measured observed using an acid-base indicator such as the juice of the red cabbage.