I improvised this mug cosy as a way to use up some of the leftover granny squares I ended up not using for this project.
I finished the granny squares off with a row of single crochet around the edges, joining to the next square as I went (instructions here). I then carried on with a row of single crochet around the whole strip, adding a chain loop for the button hole. A second row of single crochet finished the whole thing off, and made the button loop thicker.
The yarn I used to finish it off was a leftover, unlabelled skein, but I seem to remember it was a Stylecraft dk wool yarn.