How does one interpret the meaning of a stochastic derivative? My understanding of derivatives is in the difference quotient limit sense... How does one interpret the meaning of a stochastic derivative?  How can one possibly differentiate a random variable?  What is its physical meaning, if it has any?

There is some ambiguity associated with the term "stochastic derivative". It can mean "Nelson derivative" and "Malliavin derivative", for example.
The definition I use is the following:

A real-valued stochastic process $\{ X(t) \}$ is said to be stochastically differentiable (differentiable with probability one, differentiable in the $L_p$ sense) at a point $t_0$ if there exists a random variable $\eta$ such that, for $t \to t_0$,
$$ \frac{ X(t_0) - X(t) }{t_0 - t} \to \eta$$
in probability (with probability one, or in $L_p$ respectively). The random variable $\eta$ is called the stochastic derivative of the process at the point $t_0$, and is denoted $X'(t_0)$.

In other words, even though $\{X(t)\}$ may not be differentiable, there is an $\eta$ from which we can make the approximation
$$ X(t) = X(t_0) + (t_0 - t) \eta + r(t) $$
where $r(t)$ is a remainder small enough in probability or in a $L_p$ sense.
For differentiability in probabiliy, we have
$$r(t) = op(|t_0 - t|)$$
which means that for any $\epsilon > 0$,
$$\lim_{t \to t_0} P\left( \left|\frac{r(t)}{t_0 - t}\right| \ge \epsilon\right) = 0$$
