somewhere between rabbit and rare-bit

The story as I got it was that it was called Welsh rabbit because the Welsh were too poor to afford real rabbit. This was a joke since the standard method of getting (not cooking) rabbit would be to poach it. The rarebit variation would be from a misunderstanding of the joke. Since there is no rabbit in the dish it couldn't be Welsh rabbit and must have been rarebit misheard. In my youth it was a cheese/egg mixture scrambled and served over toast. The classic recipe uses beer but we were a largely non-drinking household.