Is expiation (kaffara) obligatory for someone who swore an oath that their son would not enter their house if he rented a car for travel, and then the son rented the car, it broke down a third of the way, and he returned?

It appears that you have broken your oath by your son entering the house, because you swore not to rent him the car and go with it to a certain place, and this has indeed happened. If your son entered the house, you have broken your oath and are obligated to perform an expiation (kaffarah) for the oath, which is feeding ten poor people or clothing them or freeing a slave. If one cannot find [the means], then fasting for three days, as Allah the Almighty says: "Allah will not impose blame upon you for what is unintentional in your oaths, but He will impose blame upon you for what you intended of oaths. So its expiation is the feeding of ten poor people from the average of that which you feed your [own] families or clothing them or the freeing of a slave. But whoever cannot find [the means] - then a fast of three days. That is the expiation for oaths when you have sworn. And guard your oaths. Thus does Allah make clear to you His verses that you may be grateful." If your son did not enter the house, then you have not broken your oath. However, it is better to let him enter and make expiation for your oath, for the Prophet, peace and blessings be upon him, said: "If you take an oath, and then see something else better than it, then make expiation for your oath and do that which is better."