My solid paper Collins (1872 pages ed. 2007 ) gives:

Welsh rabbit n a savoury dish consisting of melted cheese sometimes mixed with milk, seasonings, etc., on hot buttered toast. Also called:
Welsh rarebit, rarebit [ C-18th a fancyful coinage; rarebit is a later folk-etymological variant ]

This is just fun:
Windsor McCay Dreams of the Rarebit Fiend