This was happening to me so I simply went into the AIM app settings and cleared out my account information. Upon restarting the AIM app I was able to reinit my account and it is working as well as it did upon first install.
With respect to what the problem is, the crash dump AIM generated (19 of them actually) indicate that the app was killed by a signal 6. (SIGABRT)
The app is very poorly designed and I have noticed a number of problems when it is not broken. For example:
-If it is configured to not 'Sign off on exit', people can send you messages after you exit but they will never reach you or get returned to the sender.
-Yesterday I was connected from my PC and also from my iPhone 3G. Messages that I sent were delivered, but messages sent to me were only delivered to my PC. (From the iPhone it looked like I was being ignored.)
-Periodically it will start showing me 'Offline Buddies' (maybe there is a gesture that turns this on/off that I have figured out, but it is really annoying)
Does anyone out there no of a better option for chatting from iPhone (other than SMS)?