You can't do this directly from ArcGIS Desktop or Catalog but if you have administrator privileges on your computer you can get around it.
Do exactly what you would do to create a new toolbox for a map document, saving the new toolbox somewhere on your computer where administrator access is not needed.
Then just browse to the location where you saved the new toolbox and move it to the ArcGIS installation Toolboxes folder. (On my computer this is in C:\Program Files (x86)\ArcGIS\Desktop10.4\ArcToolbox\Toolboxes).
Toolboxes in this folder appear by default in new .mxd documents.
As I said, you need to be an administrator on the computer your using. Otherwise the template method may be the only way to go.
To add scripts to the toolbox, you can either do it before you move it or you need to set the permissions for the tbx file in the Toolboxes folder [Right click on the toolbox, click 'properties', then the Security tab. Click the Edit... button. Under 'Group or user names', select 'Users...' or your own user name (something other than 'SYSTEM' or 'Administrators...'. Finally, under Permissions for Users, check the Full control box, then OK out]
M.J.