Where and how did the word 'moggy' come to mean a cat?

One theory is that in Lancanshire, UK, moggy used to mean mouse and the cats were "moggy-catchers". Evetually the catcher part was dropped and so cat and mouse were referred to as moggies.

Another is it may have come from the "M" marking on a tabby's head.

http://www.pictures-of-cats.org/origin-of-word-moggy.html

In the Wigan area, moggy still means mouse. (A girlfriend of mine from Standish didn't get the point of the Liverpool joke about the moggy sitting outside the Mersey Tunnel saying, "No matter how big youse are, I'll get youse when youse come out!". She couldn't see why a mouse would be saying this outside a big hole. Wigan is about 20 miles from Liverpool. Or vice versa.) In the rest of Lancashire, moggy tends to be cat. (Source: personal experience...)