A plot could revolve around the son being sorted into a different house than expected. For example, if he's sorted into Hufflepuff instead of Slytherin like his father. He then has to deal with the disappointment of some Slytherins who expected him to carry on the family legacy in their house, while also making friends in Hufflepuff who help him see the value in his new house. He could also be on a journey to uncover the truth about his father's redemption, and in the process, he uncovers a plot by some remaining Death Eaters to use his name for their own dark purposes.
One plot idea could be that the son discovers his father's past with Voldemort and tries to make amends for it in the wizarding world. He could use his unique position as the son of a former Death Eater and a hero's friend to bridge gaps between different factions. Another idea is that he has some special magical abilities inherited from both his father and the Potter side, which he has to learn to control while dealing with the expectations of others. And he might go on a quest to find hidden secrets left behind by his father related to the Horcruxes.
The son could be a key figure in a new threat to the wizarding world. Maybe some dark magic related to his father's past is resurfacing, and he is suspected of being involved because of his lineage. However, he's actually trying to stop it. He teams up with some of Harry Potter's friends' children, and together they face this new danger. They explore old wizarding locations that were important to his father, like the cave where the Horcrux was hidden, to find clues on how to defeat this new evil.