If you edit a menu item, you want to look under "Advanced Options" and then "Page Display Options". There is a box for "Page Class" which adds a class to the page, so a body class as you wanted.
You can then tweak your template to show different pieces in CSS based on this class.
If you want to drastically alter the page, you can also assign different templates to different pages through the template manager. This takes a little more work (since you have to install and set up another template).
This would be the preferred way to handle a lot of changes so you don't have to have lots of competing CSS based on page classes.