What is the meaning of the Schwarzschild coordinate $t$ in the Schwarzschild vacuum solution inside the event horizon? According to the current interpretation of the Schwarzschild vacuum solution inside the event horizon, the Schwarzschild coordinate $r$, while becoming "the time coordinate (...) also retains its geometrical significance" (Rindler, "Relativity: special, general and cosmological", p.261, Rindler's emphasis).
Now, with $r$ in this double role, what is $t$ supposed to represent? It cannot represent time, because time is already represented by $r$; and it cannot represent any spatial degree of freedom, because these are exhausted by $r$, and by the two angular Schwarzschild coordinates.
See Box 7 in Chapter 7 of Taylor, Wheeler, Bertschinger, "Exploring Black Holes", 2nd edition, and don't forget to notice the Figure inside that Box. The authors really believe that inside the event horizon the r coordinate keeps measuring some sort of "distance to the origin".
 A: $r$ is just a time coordinate inside the event horizon. $t,θ,\phi$ are the spatial coordinates.
I assume what Rindler means by "retains its geometrical significance" is that the 3D surface at fixed $r$ is geometrically a cylinder of infinite length, whose cross sections are spheres of radius $r$, both inside and outside the horizon. But outside the horizon the metric signature on the cylinder is $-{+}+$ (it's the world-sheet of a sphere), while inside the horizon the signature is $+{+}+$ (it's a Euclidean cylinder, the cross-sectional radius of which decreases at later times (that is, smaller $r$), and goes to $0$ at the singularity).
A: 
inside the event horizon, the Schwarzschild coordinate , while becoming "the time coordinate (...) also retains its geometrical significance"

This is correct. But, …

the r coordinate keeps measuring some sort of "distance to the origin".

This is not.
The geometrical significance of the Schearzschild $r$ coordinate never was as “some sort of distance to the origin”. The geometrical significance is that for constant $t$ and $r$ the surface defined by $\theta$ and $\phi$ is a 2 sphere with area $4\pi r^2$. This geometrical significance is retained inside the horizon, and allows us to easily write the $\theta$ and $\phi$ terms of the metric, even inside the horizon.

what is  supposed to represent? It cannot represent time, because time is already represented by ; and it cannot represent any spatial degree of freedom, because these are exhausted by , and by the two angular Schwarzschild coordinates.

$t$ is indeed a spatial degree of freedom. Locally, inside the horizon you can form a tetrad with a clock and three orthogonal rulers. The tetrad can be aligned and scaled so that locally one of the rulers indicates the Schwarzschild $t$ and the clock represents the Schwarzschild $r$. Forces can be applied in the $t$ direction and in all local respects it is indeed a spatial direction.
Again, the geometrical significance that $r$ retains is not as a distance to the origin nor as a spatial degree of freedom. It is simply the relationship to the surface area of the $\theta$, $\phi$ spheres.
