Cap, you may be right on that, if the FaceBook account and all associated content are associated with her and NOT your Apple ID then she will be ok with you switching the Apple IDs.
Is the iPad she has one that came with FaceBook installed or did you install FaceBookusing yourq Apple ID? If the FaceBook app came preinstalled then she owns it and the coins...as I understand how FB works.
You can do a simple test, go to Settings > FaceBook and check how the account and password are entered. Check the Settings for FB and see if there is any tie to you.
If not, go to Settings > iCloud and Delete Account so there is not the tie there to you. Create an Apple ID and password and email address for your mother, then enter that account for her. Go to the Stores > Apple ID > Sign Out, back up and change the Apple ID to the new one you created for your mother, then move forward and Sign in...you will have to enter the new password.
Now you can test and see if she can successfully play her game.
If so, in Settings go down each app and see if any are tied to your ID and if so, change them.
That way you have not done a bulk erase of settings and content but should make the iPad hers.