I have resolved my own issue, I am sharing for others because I saw nothing in the documentation that helped, and the error messages were unhelpful.
I was updating a Field Maps Form that hadn't been updated since upgrading from Enterprise 10.9.1 to 10.2. When I made a change to one of the Field Descriptions and tried to save the form, I saw the error "Unable to save your form changes. Check the layer settings and confirm that you have the privileges required to make changes."
I found a post that suggested another field might have invalid settings, so I clicked around until I received the error, "The form element has invalid properties. Leaving the properties will discard the form element. Do you want to continue editing?" when attempting to click the next field in the list. Upon further inspection of the field properties, I saw a warning that said "List contains one or more errors".
I was left to go through trial and error. What I discovered is that we had two codes in the coded value domain attached to the field with the same description. Editing one of the descriptions to make them unique and republishing the service allowed me to make my desired form changes.
