Does anyone know why a FaceTime contact would show an incorrect facebook name? When I FaceTime my friend a totally different name comes up under the Facebook contact.
When you make a FaceTime call, the only identifying information you send is what you have got set in
Settings > FaceTime > Caller ID, which will essentially be either an email address as associated with your AppleID, or potentially (in the case of iPhones not so much iPads) your phone number.
When this information hits the recipient, it's down to their machine to extrapolate any further contact information by looking for a matching entry in their local address book, and then pulling a real contact name to replace the email address or phone number. If nothing is present, the Caller ID is passed through as is. If it finds a match, if displays that match. If it finds multiple matches, then you should get a slightly cryptic looking message that says along the lines of "Bill Gates or 2 others calling" etc.
In short, if they are seeing a weird name, it's because they have said weird name against your Caller ID in their local address book. They should search for it to fix it. My guess is that Facebook contacts syncing has gone weird.