Mostly it's to throw physics at the wall and see what sticks.
You see, there are several important energy conditions in GR. These energy conditions allow you to derive several important theorems, see here for a list.
Many of these theorems are about the topology and causal structure of spacetime. For example, assuming the averaged null energy condition allows you to rule out traversable wormholes -- which is highly desirable if you want to get rid of time machines! A very common and very cute argument goes like this: any positive energy density bends light like a convergent lens. But a wormhole does the opposite: it must, because of its topology, act as a divergent lens. So there has to be a negative energy density somewhere.
The problem, however, is that there are known physical configurations that violate some of these conditions. A famous example are squeezed vacuum states where energy fluctuations may be positive or negative. Another is the Casimir effect, where there is a constant negative energy density in the region between the plates. The problem of negative energies in squeezed vacuum states may be dealt with by observing that certain "quantum inequalities" are obeyed. They allow for short lived concentrations of energy density as long as the energy you "borrowed" is paid back later, with interest. For the Casimir effect it doesn't quite work, but the magnitude of the energy density you get is very small -- it comes with a coefficient of $\pi^2\over720$.
Both of these are quantum, but there are classical examples as well. A scalar field non-minimally coupled to gravity can produce such violations. Being classical, there is no quantum inequality to constrain it. This example is the most serious because it violates all the energy conditions mentioned above. The existence of the Casimir effect and squeezed vacuum states quickly convinced people that the pointwise conditions cannot hold and must be replaced by their "averaged" counterparts. This fixes the squeezed vacuum problem, and there are indications it fixes the Casimir effect too. But it doesn't work for the non minimally coupled scalar field example, which, while speculative, doesn't seem a priori ruled out.
That's the main crux of the issue: negative masses don't seem to be ruled out, and by all accounts they should be. If they were, we'd be able to prove that all sorts of nasty things like traversable wormholes and warp drives are impossible. Conversely, if negative masses are in fact possible, we must understand how to ensure causality holds in a universe where it's possible to make time machines.