This is mainly a British expression, now very common. However, examples have started to appear in American publications, suggesting it is making inroads there. Mad here is used in the sense of crazy or deranged, not angry.
An example was heard in the BBC Radio 4 programme Start the Week on 27 March 2006, when the presenter, Andrew Marr, described the painter Van Gogh’s mental state in this way. The Sunday Mirror mentioned that it appeared in the British film Confetti: “Naturists Michael (Robert Webb) and Joanne (Olivia Colman) want to tie the knot in the nude. They are helped by the wonderfully camp wedding planners — ‘we’re mad as cheese, us!’” Hallmark has had a line of greeting cards in the UK called Mad as Cheese since July 2003.
There are plenty of examples available online, showing it dates back to the middle 1990s at least. Logically minded readers may wonder why cheese should be less sane than any other comestible, but such considerations have no place in the creation of new expressions. You can tell that from related sayings, such as mad as a pink balloon, mad as a box of frogs, and that old Northern standby, daft as a brush.
Just how do system builders dream up names for their systems? The Matrix Prestige Xtra? That’s certainly one for the next PC Plus anthology of mad-as-cheese system names.
PC Plus, June 2005
As the most raucous audience that genteel Bournemouth can ever have entertained went as mad as cheese, you could have sworn a small black cabin at the side of the International Centre hall must have smoke billowing around.
Evening Standard, 24 Mar. 2006