Why did Prophet Noahs son not believe in Allah?

Allah showed man two ways: hidayah (guidance) and dalalah (going astray).
Allah does not put the faith into ones heart forcefully unless the person wants it using his/her own free will. Allahs bestowing the faith to someone depends on the interest and effort of the person regarding the issue.

Indeed Imam Sa'duddin at-Taftazani pointed out that fact by defining the faith as the light put into a persons heart by Allah after the person spends effort using his/her own free will. (1)

Like faith, it also applies to unbelief. If the person follows the way of unbelief and going astray with his/her own free will, and shows it with his mood and attitudes, Allah does not bestow him/her the light of faith and leaves him/her on the path of the unbelief that he/she follows.

Thus, we should evaluate the reasons for Prophet Noahs son and wife, wife of Prophet Lot and Prophet Muhammads uncle Abu Talibs unbelief, in the light of that knowledge. Though they were a wife, son or uncle of a prophet, they could not use their free will in place. They did not listen to the call of the prophets and insisted on their obstinacy. Therefore, they stayed away from the blessing of belief.

So, it depends on a persons choosing the path of belief using his/her own free will to reach real salvation and happiness. Otherwise, it is no good even if he/she is a child of a prophet.

On the other hand, Prophet Noah (pbuh) knew that his task was only to proclaim. The effect was in the hands of Allah. Therefore, with his affection as a father and prophet, he did not stop calling his son Kenan to belief and Allah until the last minute. Thinking that he may believe, in the last minute, he said to him:

O my beloved son! Come and get on the ship with us and come to faith. Do not be with the unbelievers! Look! You see the water is beginning to rise. However, despite all the offers, Kenan insisted on his obstinacy. He did not obey his father. He thought that he could manage to escape on his own. He said: No, I shall not get on! I will take shelter in a mountain instead of getting on your ship. That mountain will save me from drowning.

Prophet Noah could not stand his sons dying as an unbeliever. His last word to his son was: My son! Today there is no salvation for anyone except for the ones who attain Allahs mercy and grace with their faith and obedience. There is no point in obstinacy. Look, here the water has already surrounded us. (2)

Before Prophet Noah finished his words, a big wave entered between him and his son and swept Kenan away. Thus, obeying his soul and Satan, he received the punishment of obstinacy and unbelief right away in this world too.
References:

1. Isaratu'l-I'caz, p. 44.
2. Surah al-Hud, 42-43.