Etymonline doesn't expound the etymology and states no more than:
1785, perversion of (Welsh) rabbit, as if from rare (adj.) + bit (n.).
Does anyone have more insight into the etymology of "Welsh rarebit"?
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The origin is unknown, but its current version appears to be just an attempt to provide a term more suitable to a dish:
The first recorded reference to the dish comes from 1725, where it was called ‘Welsh rabbit’. This, historians believe, caught on as a joke, but their interpretations differ. In the late 17th and early 18th centuries, the English would often give laughably fancy names to any food or product common to a region or profession.
Use of the term Welsh rarebit began towards the end of the 18th century and became more commonplace as the dish gained in popularity. The name change was probably an attempt to make the name more fitting to the dish and drop some patronizing overtones, according to the Oxford Dictionary.