Can I ask the judge to bear the blood money (diyya) of the deceased on behalf of my son and allow it to be paid in installments over three years, given my limited income and inability to pay it in one lump sum?

Killing that results from a car accident is considered "inadvertent killing" (qatl khata'), and it entails two things: expiation (kaffarah) and blood money (diyah). The expiation is incumbent upon the killer and consists of fasting for two consecutive months for whoever does not find a believing slave to free. As for the blood money, it is incumbent upon the killer's 'aqilah (his male relatives on the paternal side). It is deferred over three years, with one-third paid at the beginning of each year. The killer is not obligated to pay any part of the blood money. If it is impossible to collect the blood money from the 'aqilah (due to their poverty or absence), the state (Bayt al-Mal) bears it. If that is also impossible, the Hanbali school of thought holds that it is waived, while some scholars, such as Ibn Qudamah and Ibn Taymiyyah, are of the view that it then becomes incumbent upon the killer. It is permissible for anyone to guarantee the blood money to the heirs of the deceased, but this guarantee does not absolve the perpetrator; rather, the right remains attached to both their liabilities.