Overriding menu items for every virtual site

I want to override the title and other menu item attributes in some virtual sites created with this module.

In other words, I create a generic menu and I use that same menu as a basis for all my virtual sites. Then I want that menu items displays some variations depending on what site are you visiting.

I guess that I would create a new form in a submodule to recollect and save into some place the variations for every virtual site. How will I save the data? How do I name every menu item (and menu item attribute) as a variable so I can call them from the template to do the override?

Would it be possible to use the "Variables" tab to set variations over menu items expressed as variables anyway?