Let $\xi$ be a random variable with $p(x)$ density function, which is like a normal distribution (i.e. $p(x)$ is increasing on $(-\infty,0]$ and decreasing on $[0,\infty)$). Denote by $s$ the maximum area of rectangle inside $p(x)$ and real line: $$ s = \max\left\{a\cdot|I| \colon a\cdot\chi_I(x)\le p(x)\right\} $$ where $\chi_I$ is the characteristic function of $I\subset\mathbb{R}$, $|I|$ is the length of interval $I$ and the maximum is taken over all possible combination of $a\in\mathbb{R},\,I\subset\mathbb{R}$ is an interval. It can be shown that $s$ can be anything from $(0,1]$.
Is there any reference to paper/book/... about this maximal rectangular area ?
Does $s$ represent a probability of some event related to $\xi$ ?
The value $s$ does not represent a probability of any event for $\xi$, but it does represent a lower bound of the probability over the same interval on which the rectangle in question was constructed. To see this, we let $\mathscr{S}$ be the set of all closed intervals in $\mathbb{R}$, so that your maximised area is:
$$s \equiv \max_{\mathcal{S} \in \mathscr{S}} A(\mathcal{S}) \quad \quad \quad A(\mathcal{S}) \equiv | \mathcal{S} | \inf_{x \in \mathcal{S}} p(x).$$
Now, defining the set $\mathcal{E} \equiv \operatorname{argmax}_{\mathcal{S} \in \mathscr{S}} A(\mathcal{S}) $ we can observe that:
$$\mathbb{P} ( \xi \in \mathcal{E} ) = \int \limits_{\mathcal{E}} p(x) dx \geqslant \inf_{x \in \mathcal{E}} p(x) \int \limits_{\mathcal{E}} dx = | \mathcal{E} | \inf_{x \in \mathcal{E}} p(x) = A( \mathcal{E} ).$$