Description
The default project toolbox is an integral part of an ArcGIS AllSource project and cannot be deleted regardless of whether or not it is used.
Solution
To use a different toolbox, add the toolbox to the project—it then shows in the Toolboxes folder—or access it from its folder location. To make a toolbox the default project toolbox, right-click the toolbox and select Make Default.
If you only want to delete the contents of the default project toolbox, you can delete them individually.