This answer appears after many, many complaints of the same thing on an Apple Forum (link included).
here is the latest answer, for which I can't take credit beyond pointing you to it and hoping it works:
Aug 8, 2014 6:13 AM
Re: can't sign in to facebook on my ipad, error message 'enter the Facebook password for "" in Settings.' in response to tracey104
I have been struggling with this error message since it first appeared on my iPhone many, many moons ago. I have tried to fix a few times but gave up trying.
I tried my Facebook password like it requested and got the "..could not communicate with server error message" like everyone else. Today it showed up again and I vowed to fix it once and for. In addition to my iPhone, I have an IPad 4. I have never seen the message appear on my Ipad. In my iPad I went into settings, Facebook, and lo and behold i saw my name right below settings. I clicked my name and it said:
Full Name XXXXX XXXXX (my first and last name)
User Name XXXX.XXXXX.33 (<<<--- this is the key to the whole mess. Firstname dot lastname dot 33)
Password ooooooooooo (this is your facebook login password)
I stared at the User Name in amazement. This is probably what the facebook setting was needing, but it was only asking for password. So here is what I did:
I hit Delete Account and it then it asked for those three pieces of information. So I entered in my iPhone what was on my Ipad 4 and it cleared everything up. Now I was lucky I had another apple device with my User Name already inputted. Where would I have found out that crucial piece of information if I had not had another apple device? It is part of your Facebook login web address. Another place you can find it is: Log into facebook account, go to About and look for your contact info, it will be something like this: email@example.com bill.smith.22 is your User Name
So basically your Facebook has three names attached to it: #1 your name = Bill Smith #2 User Name = bill.smith.22 #3 email or phone that you put in your web browser with your password to log in to facebook. Confused? So was I. I hope my instructions weren't confusing.
Please post a reply if it does work!