He is 16 years old; his father told him to drive the car, and he hit a child, and the child died. What should he do?

This person who drove the car and hit a child with it, which led to the death of the child, is not a boy, as mentioned in the question. Rather he is an individual who is accountable and is an adult; he is completely responsible for his actions because, even if we assume that the tangible signs of puberty had not appeared in his body, he is still deemed to have reached adulthood when he reached the age of fifteen years by the Hijri calendar, according to the soundest scholarly opinion. 

Shaykh Muhammad ibn Saalih al-‘Uthaymeen (may Allah have mercy on him) said: Once he has completed fifteen years, he has reached puberty. The evidence for that is the hadith of ‘Abdullah ibn ‘Umar (may Allah be pleased with him) who said: I was presented to the Prophet (blessings and peace of Allah be upon him) on the day of Uhud, when I was fourteen years old, and he did not let me join the army. According to a saheeh report narrated by al-Bayhaqi and Ibn Hibbaan: He did not think I had reached puberty. Then I was presented to him on the day of al-Khandaq when I was fifteen years old, and he let me join the army, as he decided that I had reached puberty. what matters here is the words “he decided that I had reached puberty”, because if we look only at the first report, the report of al-Bukhaari, someone may raise an objection and say that he did not let him join the army, not because he had not reached puberty and not because he had reached puberty, but because he was not qualified to fight, either because he was physically weak or for some other reason. But the report of al-Bayhaqi and Ibn Hibbaan indicates that the Prophet (blessings and peace of Allah be upon him) did not let him join the army because he had not reached puberty, and when he did reach puberty he did allow him to join the army. Naafi‘ said: I came to ‘Umar ibn ‘Abd al-‘Azeez when he was the caliph and told him about this hadith, and he said: That is the definition of puberty, and he wrote to his governors telling them to adhere to that guideline. Based on that we say: if a person has reached the age of fifteen years, then he has reached puberty and is an adult, even if he is small in physical stature and has not yet emitted semen or developed pubic hair. It may be that a person is not accountable at the beginning of the day and at the end of the day he has become accountable, if he was born at noon; when he reaches the age of fifteen years at noon, he reaches puberty and becomes accountable.

End quote from ash-Sharh al-Mumti‘ ‘ala Zaad al-Mustaqni‘ (9/296, 297) 

Secondly: 

With regard to the accident that happened, the causes and circumstances thereof should be examined. If he was at fault in his handling of the car, such as if he drove without a licence or was not of the age to drive, or he drove faster than the speed limit allowed in that place, or he was careless and did not watch the road, or he did not maintain the car properly, and the like, then he is liable for what happened. In that case he must offer expiation for that and he has to pay diyah (blood money) to the heirs of the victim, which should be paid by his male relatives on the father’s side, because he killed accidentally. 

But if there was no negligence or carelessness on his part, or any infraction of traffic rules, and he was not going faster than the speed limit, and the mistake was on the part of the child, such as if he was walking in a place where pedestrians do not usually walk, and the like, and the driver could not avoid hitting him, then in that case he is not to blame. 

It says in Fataawa al-Lajnah ad-Daa’imah:

If the driver was careless in his driving or he played a role in causing the accident, such as if he broke the traffic rules or was speeding or became drowsy and the like, or he failed to maintain the car properly and failed to make sure that it was safe to drive, then he must offer expiation for accidental killing, which is to free a believing slave. If that is not possible, then he must fast for two consecutive months as an act of repentance to Allah. But if he did not play any role at all in causing the accident, then he does not have to do anything.

End quote from Fataawa Islamiyyah (3/356) 

It says in a statement of the Islamic Fiqh Council, issued during its eighth conference in 1414 AH/1993 CE, concerning cases in which the driver does not bear any responsibility:

1 – If the accident happened as the result of something beyond his control and he could not avoid it.

2 – If the damage is caused primarily by the one to whom the damage is done.

3 – If the accident resulted from negligence or aggression on the part of another, and the other is to be deemed responsible. 

Majallat al-Majma‘ al-Fiqhi, issue no. 8, vol. 2, p. 372

Please see also the answer to question no. 52918 

We have previously discussed what is required of diyah (blood money) and kafaarah (expiation) in the case of accidental killing in fatwa no. 52809

And Allah knows best.