The coagulating value refers to the minimum concentration of an electrolyte required to coagulate (or precipitate) a colloidal sol. The greater the charge of the ion (or its ability to neutralize the charge on the colloidal particles), the more effective it is in coagulating the sol. This is explained by the charge neutralization principle, where ions from the electrolyte neutralize the charges on the sol particles.
For AgI/Ag$^+$ sol, when an electrolyte like Na$_2$S is added, the S$^{2-}$ ions neutralize the charges on the Ag$^+$ colloidal particles and cause them to precipitate. The reaction can be written as:
\[
\text{Ag}^+ (aq) + \text{S}^{2-} (aq) \rightarrow \text{Ag}_2\text{S} (s)
\]
The Na$_2$S electrolyte is the most effective because S$^{2-}$ has a high charge density, which makes it the best at neutralizing the charge on the Ag$^+$ particles.
Thus, the correct answer is Na$_2$S.