An idea I had the other day based on LCARS interfaces I've seen. An interface that interactively changes itself based on usage.
Possibly more useful for websites than for homepages on mobile devices, but I can see an application for it in both.
For mobile devices-
There are key areas that are easy to reach with a thumb for a phone. Icons are moved to this area based on how often they are accessed. Not only that, they get a background area of colour that increases or decreases in size based on usage - and it changes in colour depending on what time of day it's usually accessed (warm for day, cool for night). Less often used apps are relegated to pages further away from the home page.
Pressing on an app icon transitions to the app depending on how long it takes to load - the Adaptive User Interface (AUI) not only records when and how often apps are used, but also how long it takes to load. The longer the app takes, the further in towards the centre of the device it is, if using a "pull app tab outwards" transition.
As above, but the sweet spot for oft-used pages is different, and transition spacing is based on how long it takes for a page to load. Not only that, but the text/icon for a page can become larger or smaller based on overall usage of the website, not just the individual user's activity to give a first timer an idea of what pages folk are looking at.