Hello,
while it is not possible to add the entire 'system properties' tab to the widget properties, it is possible to add properties listed in this tab to the widget properties through modification of the widgetized web part. You can find more information in
system properties context help page. Please note this is link to localhost/KenticoCMS6 and will not work if you do not have Kentico CMS 6 installed in the standard directory. You can find the page by navigating to
CMS Site Manager -> Development -> Widgets -> {edit widget} -> System properties and clicking the question mark in the upper right corner of the tab.
Regards,
Josef Dvorak