I ran into a weird issue while working with SharePoint 2013 the other day. I wasn’t able to edit the web part properties of any web parts on wiki pages. When highlighting a web part in Edit mode, the Web Part Properties box was greyed out, and also there was no checkbox in the upper right corner of the web part to select, only a minimize button. I was also logged in as the admin so I knew it wasn’t a permissions issue.
A colleague mentioned that it might be a browser issue. I checked and I was using Internet Explorer 11. I tried the site in Chrome and everything worked fine, I could update my web part properties no problem.
Enabling compatibility mode in IE might fix the issue as well, although I didn’t try it. I do seem to recall some other issues with SharePoint using IE11. Have you run across any? Please share in the comments.