An iOS device can iMessages from iOS 5 to other iOS devices.
At iOS 6 your can sync or sue the same Apple ID on an iPhone and the iPad can then be set to send as if from the iPhone Number.
The same is true of the Messages app in OS X 10.8.2
iChat stops at version 6 in Lion.
In Mountain Lion it changes it's name to Messages.
I take it you have tried to remove the Apple ID from the iPad and then re-entered it ?
Did you turn Off iMessage on the iPad before changing the password (i.e. your were not logged in already and trying to change the password ?)
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