Moving a site in SharePoint can be tricky. There are several methods to try out.
Content and Structure or Template Create/ Use Method
http://sharepointmaven.com/how-to-move-sites-in-sharepoint/
Export / Import Method
https://technet.microsoft.com/en-us/library/ee428301(v=office.16).aspx
Backup / Restore Method
https://technet.microsoft.com/en-us/library/ee748655(v=office.16).aspx
Template Method
If it is a simple site you might be able to create a template using content and then just create site with the new template with content.
Moving a Site in Office 365
http://sharepoint.stackexchange.com/questions/180402/move-or-copy-a-o365-sharepoint-subsite
Moving a Site in SharePoint 2016
PowerShell to Copy site with content to another site collection
Moving SharePoint subsite to root of site collection using Deployment API
Rename Site Collection
Rename a Site
http://www.toddklindt.com/blog/Lists/Posts/Post.aspx?ID=372