This tweak will show you the needed steps for creating default web part containers per site.
Step 1: Create a web part container with the name (code) of the site followed by .Default like illustrated below:
Step 2: Locate the default web part and widget properties files from the following locations:
~\CMSModules\PortalEngine\UI\WebParts\Properties\WebPart_PropertiesBefore.xml
~\CMSModules\Widgets\Controls\Widget_Edit_Properties.xml
Step 3: Modify the Container field and add the following attribute:
defaultvalue="{%currentsite.sitename%}.Default"
The macro resolves the sitename of the current site when the web part and widget properties dialog are opened resulting in the selection of a default web part container: