You need to set up iChat to not log you off.

– Launch iChat
– Go to the iChat menu and choose Preferences.
– Under the General category check the option “Show status in menu bar”

This should automatically uncheck the option “When I quit iChat, set my status to Offline” This will allow her to quit iChat and still listen for IMs in the background, it also SHOULD keep her from getting logged off when the computer sleeps. She’ll also be able to control her status and messages from the menu bar by the clock.

Hope that helps.