Here's my idea of Fallout 4
Setting: West Coast
Year: 2285
You play the son/daughter of a elder in the brotherhood. Your father wants you to join the Paladins, so he sends you on a quest to find a rare gun that can destroys settelments. The reason, 'To make sure it dosen't fall in the wrong hands'
You find the gun in Vault 15.
You return to the Wastes only to find the NCR who also want the gun. They shoot you and take the gun. You return to the Brotherhood who then send paladins out to find the NCR.
You join the Paladins who are then ambushed by Ceasers Legion. You escape and return to the Brotherhood. The BoS are then attacked by the NCR. The Brotherhood win their fight against the NCR.
You are then sent on another mission to find the NCR base.
You find their base and attack them. The NCR are then destroyed. You return to the Brotherhood only for them to become hostile towards you. They have learned that your father has been telling the NCR and Ceasrer Legion about the Brortherhood and believe that you will do the same. You escape their base and head into the Wastes.
You then find Vault 13 which has been attacked. You investigate and find Super Mutants. You escape and head towards the NCR base. You tell them about what you have found and try to convince them to join you in the fight against the Supermutants. They ask you to do a few quests so they can trust you such as help them rebuild and find weaponsand ammo.
Next, you go to Ceasers Legion for help. They tell you to do a few quests such as find legion members and find some people so they can crucify them.
Finally, you go to the Brotherhood with the NCR and Ceasers Legion (who are willing to become allies for this one battle)
They ask you to so a few quests such as find dead Brotherhood members and try to find lost paladins.
When you have all three as allies, you head to Vault 13. Once there, you have a fight with the supermutants. Once Vault 13 has been cleared, you go back to the wastes. You learn that someone has been creating more mutants in the wastes. You search the world and eventully find Vault City which has now been overrun with Supermutants. You then find who has been creating the mutants, your father. You kill your father and then exit Vault city. There are still more Super mutants in the vault and with the gun you found at the begening of the game, you destoy Vault City.
However, the Brotherhood, NCR and Ceasers Legion are all at war with each other. The final mission involves you joing one of these factions and defeating the other two.
If you join the NCR, then more people join from accross the West Coast. The NCR start set up NCR bases from across the Wastes. and within fifty years, the NCR become the biggest Faction in the Wasteland. With Bases all over the country. The story of how you helped the NCR is told and is now a legend. How you killed your own father for the good of humanity.
If you Join Ceasers Legion, then the Faction will start to kill people across the wastes. NCR and BoS paladins become slaves for Ceasers Legion. In twenty years, you become the head of the Ceasers Legion.
If you join the Brotherhood of Steel, then the NCR head East due to the BoS being the most powerful faction in the West Coast.
Ceasers Legion then crumble and within ten years, only the Brotherhood exist in the West Coast.
You become and Elder and then become the head of the Brotherhood. You are said to be the best leader he Brotherhood has ever had.