I don't know the answer, but I'll take an educated guess, in case the previous suggestions don't resolve the problem.
My guess is that you have an AIM password in your keychain, but it's the wrong password, so iChat fails to log in and prompts you for a better password, while failing to update your keychain with the corrected password.
If that's the problem (or if you want to see if that's the problem), here's what you might do: Quit iChat. Launch Keychain Access (in your Applications/Utilities folder). Look under Passwords -> Application for a name beginning "AIM". If you find it, delete it. Then quite Keychain Access and launch iChat. Once again it should prompt you for the password. If it asks whether to add it to your keychain, say yes. Then Quit and re-launch iChat to see if it "stuck" this time.