brim » rim
brimming with [tears] > rimming with [tears] appears to be by far the most common eggcorn in this group. It was its appearance in the cited New York Times article that brought this group of eggcorns to my attention
brimful > rimful is barely attested.
rimmed with [tears] > brimmed with [tears] is intermediate in frequency (all quantitative claims are justified by nothing more than a Google search), and I would even have guessed that “brimmed with” was the eggcorn in this case, were it not for several late-19th-century citations in the OED (not that that’s determinative, of course).