All perfect praise be to Allah, The Lord of the Worlds.
When someone dies, the first thing to do is to pay the expenses of the funeral from his estate and if he is indebted then the debt must be paid as well, even by selling a property. This is because the Prophet (PBUH) says, "The soul of the deceased believer remains pending on account of the debt till it (the debt) is repaid."{Tirmithi}.
According to scholars of Hadith, "The soul of the deceased believer remains pending on account of the debt" means that his soul is suspended by the debt he owed. Al-Iraqi said, it is suspended in the sense that it is awaiting whether its debt will be settled or not."{Qoot al-moghtathi ala Jami`e Attirmithi, 1/326}.
Debt guarantee is a Kafala (collateral) and a collateral is undertaken on voluntary basis; thus, it is conditioned that a guarantor should be mature, adult, rational, not interdicted due to weak-mindedness, and knows the amount and type of debt. "Al-Shafie said: If a person guaranteed the debt of a deceased after he was made familiar with its amount and to whom it belongs (creditor) then he is obliged to settle that amount, whether the deceased left any wealth or not."{Mokhtasar al-Mozni, 8/207}. "It is void to guarantee what is unknown and unspecified, such as one of two debts….."{Asna al-Matalib, 2/239}.
In conclusion, if the debt guarantee undertaken by the son of the above deceased person is a true commitment on his part and meets the aforementioned conditions then he is liable for it and obliged to settle it on his father`s behalf. If his intention was to retrieve that sum from his father`s estate then it is also permissible. But, if he volunteered to guarantee that debt then he is obliged to do so and has no right to claim it from the estate. On the other hand, if he wasn`t truly committed to settle his father`s debt in the sense that he wasn`t aware of its type or amount then he isn`t obliged to settle it, but it is recommended that he does in order to free his father from the liability before Allah the Exalted. This is attested to by the following narration: Narrated Salama bin Al-Akwa`: A dead person was brought to the Prophet (PBUH) so that he might lead the funeral prayer for him. He asked, "Is he in debt?" When the people replied in the negative, he led the funeral prayer. Another dead person was brought and he asked, "Is he in debt?" They said, "Yes." He (refused to lead the prayer and) said, "Lead the prayer of your friend." Abu Qatada said, "O Allah's Messenger (PBUH)! I undertake to pay his debt." Allah's Messenger (PBUH) then led his funeral prayer."{Bukhari]. And Allah the Almighty knows best.