Can we override Drupal includes files? I want to modify the menu.inc file that is in the Drupal "includes" folder, but I want to create an overriding file for it, so that we can easily package only our custom theme.
Generally Clive is right. Don't touch core. But sometimes you are forced to do a core patch or it's just much easier to solve your requirements.

If you must: I'd recommend to put your whole project into git. This gives you the possibility to have a clean core version in one branch and the patched one in an other. Everytime a new core version is published you can merge it in a clean way.