The maximum area is $3 \sqrt{5}$.

We deduce from the area of $P_1P_3P_5P_7$ that the radius of the circle is $\sqrt{5/2}$. An easy calculation using the Pythagorean Theorem then shows that the rectangle $P_2P_4P_6P_8$ has sides $\sqrt{2}$ and $2\sqrt{2}$.
For notational ease, denote the area of a polygon by putting brackets around the name of the polygon.

By symmetry, the area of the octagon can be expressed as
$$[P_2P_4P_6P_8] + 2[P_2P_3P_4] + 2[P_4P_5P_6].$$
Note that $[P_2P_3P_4]$ is $\sqrt{2}$ times the distance from $P_3$ to $P_2P_4$, which is maximized when $P_3$ lies on the midpoint of arc $P_2P_4$; similarly, $[P_4P_5P_6]$ is $\sqrt{2}/2$ times the distance from $P_5$ to $P_4P_6$, which is maximized when $P_5$ lies on the midpoint of arc $P_4P_6$. Thus the area of the octagon is maximized when $P_3$ is the midpoint of arc $P_2P_4$ and $P_5$ is the midpoint of arc $P_4P_6$. In this case, it is easy to calculate that $[P_2P_3P_4] = \sqrt{5}-1$ and $[P_4P_5P_6] = \sqrt{5}/2-1$, and so the area of the octagon is $3\sqrt{5}$.