Hi,
Your specs say OS X 10.8.2 which would mean Messages rather that iChat.
Messages gives you the iMessages account to use with your "registered" Apple ID.
You can of course have AIM and Jabber Screen Name and IDs set up as well.
I would start with using the Disk Utility (Applications/Utilities) and it's First Aid option to Repair Permissions.
I presume when the app does start up it is showing at least one "active" chat in the Messages window ?
DO you mean you are using the "X" that shows up when you mouse over the name of the person in the Chat ? (which Closes/removes the chat from the window)
Do you mean you are using the File Menu option to Delete the Conversation ?
Is it an iMessages or other form of conversation ?
7:19 PM Wednesday; November 14, 2012
Please, if posting Logs, do not post any Log info after the line "Binary Images for iChat"
iMac 2.5Ghz 5i 2011 (Mountain Lion 10.8.2)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
Couple of iPhones and an iPad
"Limit the Logs to the Bits above Binary Images." No, Seriously