Note: This solution doesn't automatically set the current menu and submenu item. If you want to highlight a particular menu as current when the hidden page is viewed, see my other answer.
From the answers that come before me, you can see that there are many ways to do this. However, there is another way that I think may be the best.
Loading the page differently based on the value of a $_GET query var is one option, but it may not be what some people are looking for.
The suggestions regarding add_submenu_page() are on the right track, but each of the previous suggestions have problems. Setting $menu_title to null doesn't keep the menu item from being displayed, it just makes it so the link doesn't have any text. The link still takes up some room in the menu though, so it looks funny. Setting the $parent_slug to null doesn't have this problem, but I noticed that the page's HTML title doesn't display the $page_title text.
My solution was to set $parent_slug to a fake menu slug, like 'i_dont_exist'. The menu item won't be displayed, and when viewing the admin screen the page title will be filled out properly.
add_submenu_page(
'_doesnt_exist'
,__( 'Page title', 'textdomain' )
,''
,'manage_options'
,'menu_slug'
,'display_my_menu'
);