Praise be to Allah.
Rent-to-own schemes take several forms, some of which are permissible and some of which are disallowed, as is explained in the answer to question no. 97625.
It is a rental contract, until ownership passes to the renter, either by selling the item to him or giving it to him.
It cannot be said of the son in this case that he is a purchaser, rather he is a renter.
If the father rented the car for the son, and all that happened is that he put the contract in his name, then the debt is owed by the son, and he must pay it off, and the fact that the bank did not ask for it due to some problem in its system does not excuse him from that.
If the son cannot pay off the debt, then he should go to the bank, cancel the contract and return the car to them.
If the bank waived the installments from the father because of his death – as is done by some banks – then the son must go to the bank and explain to them that he is the real renter.
Shaykh Sulaymaan al-Maajid (may Allah preserve him) was asked: O shaykh – may Allah preserve you and take care of you – I have a question. Muhammad agreed with his brother Fahd that Muhammad would borrow 100,000 from the ar-Raajihi bank and give the money to Fahd, who would repay it. Then Muhammad died. What is Fahd required to do for the bank or his brother, if the bank waives the loan for his brother?
He replied: Praise be to Allah. To proceed: So long as Fahd is the real borrower and beneficiary, then he must pay the installments [on the loan] to the bank, even if his brother has died, and he must inform the bank of that. And Allah knows best
End quote from the shaykh’s website.
If the son discovered a defect in the car during the rental stage, or during the purchase stage after the rental had ended, then he may go back to the company and ask for a reduction in the rental or in the price, or for an annulment of the contract.
It says in Kashshaaf al-Qinaa‘ (3/218): Section on one who purchases a defective item and was unaware of the defect at the time of the transaction, then he found out about it. He has the option (of annulling the deal), whether the seller was aware of the defect and concealed it from the purchaser, or the seller was unaware of it. … The purchaser has the option of returning it because of the defect, so that he will not incur loss if he keeps it with this defect.
If he returns it, he may take back the price in full, because by annulling the deal, the purchaser has the right to take back the entire price… Or he may keep the item, with some compensation for the defect.
And Allah knows best.