I have a publishing site. A user is in the visitors group. On a certain page I have given the user Full control. They can edit the body of the page but cannot edit the content of a ContentEditor WebPart. I can't work out what the permissions needed are when the user has full control on the page.
If you add CEWP to a page, and then sign in under that user you may see label under CEWP - "You do not have the Add and Customize Pages permission, which is required to edit the contents of this web part.". It doesn't matter what permissions user have to page, or library, but he must have "Add and Customize Pages" site level permission.
I tried to get full control to page, to library, even to MSContentEditor.dwp file in web parts gallery, but without success. Only get user "Add and Customize Pages" helped. I faced with this problem in past, and even have find some explanation on msdn forums, but sorry, I lost the link.
When you say "cannot edit" what exactly does the user experience? If the user has Contribute or greater permissions to the page then they can interact with all of the web parts. It's been my experience that users are often confused by the UX around content editors and can't get past the "Modify Shared Web Part > Rich Text Editor" flow. They expect to be able to edit the text in-place in the CEWP as soon as they click Modify Shared Web Part.
If you are using Sharepoint 2010 . There are few things changed in respect to permission Check this blog for the details Link
I have found additional limits to these settings. In our SPO environment, the user must have at least Design permissions at the site level. If the site level permissions are Edit or below, the Content/Script Editor Web Parts still cannot be edited, even with Full Control permission in the library. (even with the Admin Center scripting option set to allow, and the user is in the Style Resource Readers group)