Here's just and idea (feel free to use it in any way):
Imagine having a network of small (4-6 passengers) vehicles servicing a city for daily transportation needs. Users would enter a desired location and arrival time. The arrival time could be flexible (within an hour, if not then the price could be appropriately higher) and the user would announce any regularity (for example detailing a weekly commute) that could be used for future planning.
The centralized system would optimize the problem of getting all passengers to their respective locations and suggest departure time and location (preferably within a few minutes of walking distance).
An interesting open source implementation would use OpenStreetMap data and have simulations and visualizations. A commercial entity could deal with deployments on various locations and provide a stable software as a service around the core open implementation. Autonomous vehicles would provide much more efficient operation of such a network and lower the costs significantly.