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I know that $\ce{CO2}$ can dissociate into bicarbonate ions within $\ce{H2O}$. If Na is added, then sodium bicarbonate is produced.

Could you then create sodium bicarbonate using carbonated mineral water and sodium in the form of table salt or does that have too many minerals aside from $\ce{H2O}$ in it?

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Assuming $\ce{Na}$ is used to denote sodium metal and $\ce{NaCl}$ is used to denote table salt, the following reactions in water occur:

\begin{align} \ce{Na + 2H2O &-> 2Na+ + 2OH- + H2(g)} \tag {1}\\ \ce{NaCl(s) &-> Na+ + Cl-} \tag{2}\\ \ce{CO2_(g) <=> CO2(aq) &<=> H2CO3(aq) <=> H+ + HCO3-} \tag{3}\\ \ce{H+ + OH- &-> H2O} \tag{4} \end{align}

So (1), (3), and (4) would give you sodium bicarboante if you dried the solution.

(2) and (3) would give you just sodium chloride back if you dried the solution.

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