You remember the part in the story where heifer-Io is tortured by Hera's gadfly, right? Well, in case you aren't up to date on your mythological bugs, a gadfly is a stinging insect (kind of like a horsefly) that likes to buzz around bug the crap out of livestock. (Nasty, huh?) These days, though, you can use the word "gadfly" to describe more than just an annoying bug. The word can also be used to describe an annoying person, especially someone who annoys with lots of constant "stinging" criticism.
We should also point out that this isn't the only time that gadflies pop up in Greek mythology. Hera also sic-ed one on Heracles when he was stealing the cattle of Geryon, and Zeus sent one to sting Pegasus when Bellerophon was trying to ride the flying horse up to Olympus. So you could say that throughout Greek mythology, gadflies symbolize the wrath of the gods and the power they have to punish mortals who get too big for their britches.
Insects symbolize the anger of the gods in other religions as well. Just check out the nasty plague of locusts that the Judeo-Christian God sends in to torture the Egyptians in the book of Exodus. Yikes. That sounds way worse than one annoying gadfly.