The idea may be plausible, but it comes with a big problem: it can't be tested. Mass is what’s known as a dimensional quantity, and can be measured only relative to something else. For instance, every mass on Earth is ultimately determined relative to a kilogram standard that sits in a vault on the outskirts of Paris, at the International Bureau of Weights and Measures. If the mass of everything — including the official kilogramme — has been growing proportionally over time, there could be no way to find out.
The way to measure the increase in weight is actually described in the article itself: observe a shift in absorption/emission spectra.
Of course, such a shift may be infinitesimally small and undetectable over short timespans. However, this is precisely the sort of theory (like all the neutrino theories, for example) for which you set up a very long-term experiment.
The experiment is simple: measure the absorption spectrum of hydrogen in space with an extreme level of accuracy and a bajillion measurements (that's the scientific term). Record that for posterity. In 10 years (the equivalent of observing something 10 light-years away), repeat the measurement. Keep doing this until you either observe a shift or have made a measurement that based on observations of the real world (i.e. galaxies drifting away).
The experiment might take 100, 1000 or even 10'000 years, but eventually we'd have an answer!
UPDATE: Heh. isomorphic points out that this is actually not feasible given that it's impossible to make these measurements accurately without relying on mass being constant... So much for this. retracts