Browsing through menu related stuff in the DOM lead me to nothing, but Google turned up this blog entry by Harbs: http://in-tools.com/article/scripts-blog/showing-text-formatting-overrides/
-- and then I could find it in the DOM, well hidden:
Adobe InDesign CS6 (8.0) Object Model JS: ScriptMenuAction
So it seems you have to set the state of a menu action. The action in itself can be attached to one or more menus (MenuAction, where checked is read-only, is a parent class of ScriptMenuAction, and there it is read/write), probably to ensure that if a certain menu item appears in more than a single menu, they are all "updated" at the same time.
Harbs' example always sets 'checked = false' and changes the menu text to indicate its current state, but I assume one can also toggle the checked state.