Could you theoretically map the internal distribution of mass in a black hole using Hawking radiation? Assuming you could measure the qualities of the radiation emanating from all around a black hole, could this be used to determine the internal geometry or makeup of the mass inside?
 A: In principle yes. But it is very, very hard:
Computational Complexity and Black Hole Horizons
Quote from the paper:

Representing bulk physics by boundary data at a ﬁxed time is diﬃcult
  for well known reasons. It is especially diﬃcult to represent the bulk
  degrees of freedom behind the horizon. And ﬁnally it is
  extraordinarily diﬃcult to represent them at late time. The diﬃculty
  is connected with complexity and chaos. The purpose of this section is
  to explain these diﬃculties.

The conclusion of that section is, quote:

The bad news of this section is that chaos and complexity make it
  impractical to compute the gauge theory description of even a simple
  low energy observable behind Bob’s horizon at late time. The good news
  is that the diﬃculties are practical and do not present an
  in-principle obstacle to describing the interior

A: The answer is no, if our current understanding of Hawking radiation is correct. The problem is that Hawking radiation is thermal. This means that it comes in a statistical ensemble of quantum states. In particular it can't carry information about the evolution of quantum states you dropped into the black hole. This is the origin of the famous information paradox. 
Of course nobody has solved the information paradox, although several competing explanations have been advocated. It's conceivable that perhaps there is hidden information in Hawking radiation within a more complete understanding. 
