I finally figured this out.
The solution was to go to the Contacts app and change my name there.
My contact -- the one with my name -- had my AIM addresses listed there. Messages must be looking up the AIM address in the contacts list and taking the full name from there. When I change my name there, it changed in Messages.
I have no idea why this is what did it, but it worked!
Messages and iChat before that is closely linked with the Contacts app (Formerly the Address Book)
When you Add A Buddy you can add the First Name, Last Name and Create an entry (Address Card) that the Contacts app uses. (You can also link to an Existing card)
In the Bonjour Buddy List Messages uses your My Card from the Contacts app (Head and Shoulders icon) and Broadcasts this as your Buddy name to be displayed on other people Bonjour Lists.
As you have found the top of th Buddy Lists can display the Handle (AIM screen Name or Jabber ID) or the Name from the Contacts list > My Card.
Other related items
With the Messages window with a chat open go to the View Menu > Buddy Names
You can use Full Names, Short Names (Nicknames or First names) and handles here as well.
If you have a Buddy List "Front" the some Menu and option changes the Buddies in the same way.
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