OK, go to Security->Access Controls -> Assess policies tab.
Right-click on the Administrator policy and select edit.
Click on the Permissions tab.
Click on the New Permission button.
Put support_menu in the "Name" field.
(The name is arbitrary, but it has to match exactly what you use in the Permission field below.)
Put "Show the Support Menu" in the Description field.
Click on "Add"
Click on "Save"
That gives you (as the admin) that permission.
Now, go to System - > Actions
In the "Top Menu" panel (right side) right-click on "Support"
Select Update Menu
put support_menu in the Permissions field.
Click on Save.
Now a user needs that permission to see the Support Menu. Viola.
For some other menu choices, there is already a permission in the Permission field. If you want to hide those, just add a comma and your own permission at the end (no spaces).
Note that your change to the Administrator policy could be overwritten during some upgrade installs (I’m not sure). That would make the Support menu disappear for you (the admin) and you’d have to edit the DB to get it back. To avoid that, you could duplicate the Administrator policy and change all policy references in all ACL entries from Administrator to the duplicate policy.