Certain themes come with a home page slider that either displays the latest posts or can be customized using the Slider admin page.
On the Slider admin page, you have the ability to search for a post/page and add it as a custom slide. Each custom slide can be edited to replace the default title, image, excerpt and link. You can even reorganize the slide orders by dragging them around.
If you just want to have the slider display the latest posts, but want to use a custom image instead of the featured image you can easily set the custom image on the post edit screen.
Clicking on the “Set Image” button will bring up the Media Uploader where you can select or upload an image.
NOTE: Not all themes comes with a home page slider and certain themes, such as Tango, will only work using custom slides.