It isn't clear exactly what sort of initial conditions you're requiring.
The difficulty with minimal initial conditions is part of the reason why it was an amazing result when Huisken--Ilmanen constructed a "weak inverse mean curvature flow" which
(1) Can start at a minimal surface (technically, for certain things to work nicely, it should be outer-minimizing)
(2) Exists for all time in an asymptotically flat manifold.
AND
(3) Still satisfies Geroch monotonicity, i.e. the Hawking mass is monotone along the flow.
That one could find a "flow" which satisfies (1) and (2) while still keeping (3) is incredible.
Their paper is very readable, although quite long, so I'd recommend that you take a look at it, rather than I try to explain the ideas here.
EDIT: I've added some more information below. Its not totally clear what your motivation for the question is; if you add more information perhaps I can answer your question better.
It is sometimes possible to define a classical flow which starts at a minimal surface. For example, in the Riemannian Schwarzschild metric $$ g = \left(1+\frac{2m}{r}\right)^4 \delta, $$ on $\mathbb{R}^3\setminus \{0\}$, there is a inverse mean curvauture flow $\Sigma_t = \{r(t)\}\times \mathbb{S}^2$ defined for $t>0$, where $\lim_{t\searrow 0} r(t) = m/2$. I'll leave it to you to compute the associated ODE (HINT: the easiest way is to use the fact that $|\Sigma_t|=e^t|\Sigma_0|$.
Be very careful with what I mean here for $t=0$. In particular, the PDE is not satisfied for $t=0$, just $t>0$.
Here are some results about the classical flow:
http://www.ams.org/mathscinet-getitem?mr=1064876