Spoilers for the Main story line up ahead for all those who haven't finished the game yet!
What if after the completion of the main quest, if you have taken Synth-Shaun in as your own and providing his story isn't continued canonically by future DLC, you are able to begin a quest that has you head either into The Institute/Institute ruins (obviously depending on choices) and uploading Shaun's software into an adult Synth body. The model could be made to still look somewhat young, like maybe 18-20 years old, to keep the child-parent dynamic between the him and the player. Someone interested in making a follower mod to rival Inigo for Skyrim or Willow for new Vegas could voice the adult Shaun and i'm sure there's enough lines of dialogue for the player to piece together decent father-son conversations.
Could even make him non-essential and see how travelling the wasteland with a son you must protect affects people's play styles.
I know I'd enjoy it, i think its a great jumping off point to have you make an emotional bond with a character like I know a lot of people did with Inigo. That being said, optional backstory aside Inigo was a total stranger and it can be more fun for potential follower creators to start an original character.
Although, an additional idea giving modders the opportunity to breathe whatever kind of life they wanted into a basically blank slate would be to have a mini quest when synth-Shaun is still a child to perhaps take him a few places and face a few moral dilemmas and your choice affect how his personality turns out once aged up. This teaching process could even continue after that, a chance for your character to "do it right" after having your son turn into the power man institute overlord previously. Or even to ensure he follows in original Shaun's foot-steps and becomes the perfect heir for when you pass over the helm of the institute. obviously all depending on your in-game choices. though i'm aware that's a lot of work for modders and voice actors alike with multiple personalities to create. Just a thought, feel free to let me know what you think.