Adding a favicon to your landing page is easy. Go to the SEO section of the Experience Manager for the experience you want to change the favicon of, and then just click on the Upload button next to the favicon to add your own.
You can use .jpg and .png images (.ico is not currently supported). You can also use .gif files, but some browsers might only display the first frame of the GIF, without animation. Make sure the image used is not too big, the recommended size is 32 by 32 pixels.
Don't forget to save your changes and update the experience afterwards!