Variation of the metric with respect to the metric

Since the metric $g_{\mu\nu}=g_{\nu\mu}$ is symmetric, we must demand that

$$\begin{align} \delta g_{\mu\nu}~=~&\delta g_{\nu\mu}\cr~=~&\frac{1}{2}\left(\delta g_{\mu\nu}+\delta g_{\nu\mu}\right)\cr~=~&\frac{1}{2}\left( \delta_{\mu}^{\alpha}\delta_{\nu}^{\beta} + \delta_{\nu}^{\alpha}\delta_{\mu}^{\beta}\right)\delta g_{\alpha\beta},\end{align}\tag{1}$$

and therefore

$$ \frac{\delta g_{\mu\nu}}{\delta g_{\alpha\beta}} ~=~\frac{1}{2}\left( \delta_{\mu}^{\alpha}\delta_{\nu}^{\beta} + \delta_{\nu}^{\alpha}\delta_{\mu}^{\beta}\right).\tag{2}$$

The price we pay to treat the matrix entries $g_{\alpha\beta}$ as $n^2$ independent variables (as opposed to $\frac{n(n+1)}{2}$ symmetric elements) is that there appears a half in the off-diagonal variations.

Another check of the formalism is that the RHS and LHS of eq. (2) should be idempotents because of the chain rule. For further motivation, see e.g. this Phys.SE post.