It may have been a deliberate attempt to “glamourize” the dish.
It may have been suggested in humorous vein; or it may have been a failure to understand the country humor in the original name, but whatever the reason, the word rarebit is used only to designate the cheese dish, Welsh rabbit, normally also in combination, as Welsh rarebit.
Similar terms originating in popular humor are Adam’s ale (water), Scotch coffee (hot water flavored with burnt biscuit), Missouri meerschaum (corncob pipe), Cape Cod turkey (codfish), and so on.