Q: How can I arrange for the visibility of certain menu items to change based on the logged-in user?
A: Use the most excellent 'Menu per Role' module and then configure your menu item's settings in the 'Restrict access permission' section.

If you want to restrict certain menu options for a group of users first add a new user role (/admin/user/roles) and add the users in question to it (if you haven't already!). Now install the 'Menu per Role' module, this will allow you to configure menu premissions on a user role basis (and therefore on a user basis).

To configure a menu item's permissions, edit its settings, and in the 'Restrict access permission' section select the user roles that should have accss to the menu option - once configured other users will never see the menu item at all.

By default all roles have access to all of the menus so you only have to update the menu items from which you want to restrict to certain roles and users.