Why does one run around like a blue-arsed fly?

Seems like a bit of a mixed metaphor. Run around is a common expression meaning to be busy, frantic, or otherwise getting the runaround. But to emphasize how much one is running around, we turn to metaphor. The fly in question is apparently a blue bottle fly, which buzzes around rather frantically, like someone who is running around busily doing errands.

A quick look at Yahoo Answers gave me this:

Certainly the fly in question is a bluebottle, but I think I'm right in saying that the expression originally was "buzzing around like a blue-arsed fly", which makes a lot more sense. One more normally tends to "run around like a headless chicken".

Sounds plausible...

If you watch a bluebottle, the fly in question, they buzz around in a generally hectic and irritating fashion without - as far as one can tell - getting much done.

If one is running around like a blue-arsed fly you are not running around in the same way the fly would run around, but you are running around in the way the fly will fly around- hectic, hurried, noisy, maybe a little annoying and typically not - as far as one can tell - getting much done.

When I was young we often referred to the bluebottle as a 'blue house fly'. It is most likely that the modern expression of 'blue-arsed fly' is corrupted from this expression. Such mutations or derivations of expressions based on how they sound is, of course, not uncommon. The problem is producing evidence to back this up from an era before everyone told you what they just had for breakfast through social media!