In GR what is a "source-frame mass"? In the recent LIGO paper, 
"GW150914: First results from the search for binary black hole coalescence with Advanced LIGO, (LIGO Scientific Collaboration and Virgo Collaboration)", 
they refer to the masses in some models as:

"... a nearly equal mass black-hole binary system of source-frame masses $36^{+5}_{−4} M_{\odot}$ and $29^{+4}_{−4} M_{\odot}$ (median and
  90% credible range)."
"... a stellar mass binary black hole system with source-frame component masses $23^{+18}_{-5} M_{\odot}$ and $13^{+4}_{-5}M_{\odot}$"

In General Relativity I've heard of different types of mass parameters such as Komar, Bondi, or ADM mass.  But what is the source-frame mass? How are they getting a localized measure of mass in such a dynamic spacetime?
 A: It's just an effect of living in an expanding universe, just as wavelengths from a source will be stretched by a factor of $1+z$ for redshift $z$ between when they are emitted and when we receive them.
As described in this paper (the reference after your first quote),

The observed frequency of the signal is redshifted by a factor of $(1+z)$, where $z$ is the cosmological redshift. There is no intrinsic mass or length scale in vacuum general relativity, and the dimensionless quantity that incorporates frequency is $fGm/c^3$. Consequently, a redshifting of frequency is indistinguishable from a rescaling of the masses by the same factor. We therefore measure redshifted  masses $m$, which are related to source frame masses by $m = (1+z) m^\text{source}$.

That is, the naive mass measurement assuming a Minkowski universe is going to be wrong by a factor of $1+z$ compared to being right next to the black holes and weighing them. The team corrects for this given the redshift of $z = 0.09$.
As for what masses these are, you're right things can get confusing in general relativity, especially when things are interacting. However, isolated black holes sitting in asymptotically flat backgrounds do have well-defined masses everyone agrees upon. The numerical simulations are able to say what these masses are (essentially the masses of the well-separated components at the initialization of the simulation, and the mass of the remnant at the end, long after everything has settled down), and that is what is being reported. No one is trying to partition the total ADM mass between the black holes and their nonlinear interactions halfway through the merger.
