Another really cool use case will be sending your car to drive around the block for an hour or go back to your house and park, while you are going out to places. Beats paying the $10 to park for an hour.
Lots of interesting effects there, all the way up to basic urban land-use decisions.
Suddenly the price people will pay for parking is going to be closely related to the price of gas. The maximum value of a parking space in the city is basically the cost to send the car back out to some suburban parking garage during the day and have it come back in later for you. My guess is that the market-clearing price is going to be a lot lower than it is today. Lots of urban parking structures might end up getting repurposed for other uses, if you can't fill them at hundreds of dollars per space per month.
In the short run I could see the average self-driving car's gas usage being higher than a conventional car's, but in the longer run the same self-driving features might make fully electric cars more palatable. Long charging times don't really matter if you can let the car go and charge itself while you're at work, for e.g.
And if that place that it goes to charge itself happens to have a pile of solar and wind, even better! Way easier to do a bit installation in a less populated area than right in the centre of downtown.