When adding a web part to a Web Part page, the only option I have are Lists and Libraries category. I'm looking to add a Content Editor web part, but that is not available. Is there a setting under Site Administration to change this?
To enable Content Editor web part you should activate BasicWebParts feature on that site.
You can do this using following PowerShell script.
Enable-SPFeature 00bfea71-1c5e-4a24-b310-ba51c3eb7a57 -Url http://siteurl
There is a catalogpage for webparts in
your site URL here
/_catalogs/wp/Forms/AllItems.aspx click new and check the box preceding
Microsoft.SharePoint.WebPartPages.ContentEditorWebPart click the
populate button to commit your changes.