This idea is a solution to one of the parameters from Elon Musk's Master Plan, Part Deux. The plan mentions "Enable your car to make money for you when you aren't using it".
When the car is not in use, it could explore and collect data on the areas in the city or the outskirts that have not been explored. It could record the speed it can achieve in these areas, the signals, congestions, and roadblocks/ repairs, and send data back to the company. The exploration could also suggest where you need charging stations.
Data generation for self-driven car companies
Extra money for the car owners
The car company benefits because they can study the car. For an example, damage caused on certain terrains and how to prevent future damages.
The company can also sell the videos/photos the car captured of magical destinations to content creators.
You initiate "exploration" when you know you will not be needing the car for atleast the next 6 hours. You can do this every night. However, if you do it during the day, data on traffic congestion could also be gathered and the owner could earn extra. The self-driven car then receives an exploration route from the company and it starts exploring. You get the real-time location of the car throughout the exploration. Even if all the areas are explored, there are changes happening like construction, roadblocks, etc., which could be detected during exploration and sent to the company. The next day, your car will avoid those routes. Several cars could be logged in and explore the entire city. Since the data is shared across the system, you benefit from someone else exploring the previous night.
Benefits other than money for the owner may include free servicing, free repair if the car is damaged during exploration. As Goran Radanovic mentioned, lifetime free service and repair as compensation, a special discount for owners to trade in those cars for new ones are other benefits. What else?
One major problem is these cars unnecessarily increasing the traffic on the roads.
What if the owner has an emergency and needs the car back before completing the exploration? May be owner gets a free cab service during the emergency period.