I want to be able to go invisible in iChat, like I can via Gmail. How can I do this?
Is is possible to go invisible in iChat
ichat
Related Solutions
Your problem might be your router loosing connection temporarily to your ISP.
I use Chax, an extension to iChat which, among other great things, auto-reconnects when connection is lost. Chax is donationware.
I actually just tried this and iChat does it automatically. As you can see I have a different window for each account, and I think you can actually set the window's title by setting the account description under Settings->Accounts. As you can see I even have different groups for each account. 
Although I usually have them unified using Chax's feature Use All Contacts window as primary contact list
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Best Answer
When using Gmail in iChat before 10.7 you can't go invisible. I'm using Adium to be able to go invisible when I login via Gmail.