Current name is meaningless if you haven't heard the quote it comes from.
Possible to mistake for an unrelated meaning.
The quote/historical event is allegedly obscure.
Quote comes from a situation that used the trope to a T.
The quote/historical event is allegedly well-known.
Journalists appear to like twisting this quote around to apply to other things, mainly through the use of snowclones. This shows that the quote is well known (amongst journalists at least), though not necessarily known for this context.
"Will rid me of this" returns 107,000 hits on google. The first one that isn't about Henry and Becket is a quote from Bill O'Reilly riffing on it ("Who will rid me of this troublesome Doctor?") at #7. The top results are split between actual invoking of the quote in similar situations, references to Henry/Becket and works about them and unrelated snowclones.