To allow yourself to be able to copy your away message, add your own AIM screen name to your buddy list. Then, view your profile by either pressing Command-G and then type in your screenname or by Control-clicking on your name and selecting Show Member Information from the pop-up menu.
[robg adds: I'm not sure why you'd need to copy your away message, given how short most of them are, but if you need to do so, here's a solution. I haven't tested this as I don't have AIM installed...]

