from Dr. Bill [wwh]:

Instead of having two cups of coffee, I
like to have it in a mug, which holds a bit over 2 cups.
It occurred to me to wonder how they got the name.

from online etymology dictionary:
mug (n.1)
"drinking vessel," 1570, "bowl, pot, jug," perhaps from Scand. (cf. Swed. mugg "mug, jug," Norw. mugge "pitcher, open can for warm drinks"), or Low Ger. mokke, mukke "mug."
mug (n.2)
"a person's face," 1708, possibly from mug (n.1), on notion of drinking mugs shaped like grotesque faces. Sense of "portrait or photograph in police records (e.g. mug shot, 1950) had emerged by 1887. Verb sense of "make exaggerated facial expressions" is from 1855, originally theatrical slang.

Here's a link with pictures of Toby mugs. They were
dicussed so long ago, few members are likely to remember.

http://pages.heartofohioantiques.com/
995/InventoryPage/541/1.html