Considering Skyrim's Honorhall and the Aventino/Dark Brotherhood quest that started that game's adoption system in Hearthfire, I'd like to see something similar, yet more unique to Fallout 4.
Between the player being able to build up settlements and the faction system, I can see a mission dealing with the player helping to create an orphanage which function relates to the faction the player focuses on in the game. Such as Brother of Steel recruitment and training facility, a method of seeding the Commonwealth with Synth 'children' (according to Celtic belief that faeries try to swap human children with faery children which were called Changelings; this would be a 23rd Century version) by the Institute, to the Minutemen trying to save those left to fend for themselves.
It could start out with Nannybot with a couple of children asking for help from the player to find a safe place for the robot to better take care of/protect the kids to starting up an orphanage, which could lead to deciding which faction should take over.
The player could send orphans or take them to the orphanage. Maybe they could be asked to do missions to resupply the orphanage, or escort potential parents to the orphanage. I could say that these missions would include having to defend the orphanage, but I think the system needs fixing in regards to defending settlements in FO4 first.