- Go to the appropriate page
- Append ?contents=1 in the URL right after .aspx
- Press <Enter>
- SharePoint displays the Web Part Page Maintenance page
- Select the web part of your interest and then click the appropriate funciton on the toolbar, such as Delete
Keep in mind that whatever changes you make on the Web Part Maintenance page will affect all users who have access to the actual page.
One last thing. There's a reason why I always recommend my customers to always rename their web parts immediately after they have added them to a page. And the reason being is the Web Part Page Maintenance page. Once you are in this page, there is no way to distinguish a web part from another one if they are all called Content Editor, Image Viewer, or Calendar. I don't know you, but I'm not a mind reader.
If you cannot distinguish one Content Editor Web Part from another one because your customer failed to rename them after adding them to a page, try to rename the other web parts that are not corrupted. This will help you narrow it down. If that's not an option, well, I'll leave it up to you on how to disclose the bad news to your customers that you may end up removing the wrong web part. Believe me, I've been there myself and once I removed the wrong web part (or all of them because there was no way for me to tell them apart from this page) my customers learned their lesson and never forgot to rename their web parts ever again.