PvP
Raiders and Radroaches are a joke when you equip you dwellers with powerful weapons. So the option to invade other vaults could be a nice addition. Prizes could range from caps to lunchboxes to weapons to rare dwellers. Of course this could be a problem as the game is an offline game, however that could be easily solved by having it only possible to attack or be attacked while online. However, if this is made possible, I would like to request that dwellers that die in attacks/defenses come back to life for free after said attack/defense is over as losing a high level dweller in an attack gone wrong would be highly unfair. Also balancing needs to be a factor.
Move/Rebuild Funtion
Currently in Fallout Shelter the only way to move a room is to move you dwellers out of the room, sell the room, rebuild the room in another location, move dwellers back in. That takes up way to much caps, so unless you pre plan, you are left with a pretty ugly looking vault. A move function would allow you to pause time and move rooms around freely without having to waste any caps. A side tab could also be used as a way to store rooms that you are currently not moving in order to not clutter up the screen. The store could also be useable.
Family Info Bar/Family Tree
Families in the game are hard to keep track of. So my suggestion is a info bar on the dweller stats so know who is related to who and how. That way I would know who I can breed together and stop putting family members in the barracks together and wondering why nothing is happening. Another feature that could also help would be a Family Tree which shows the different generations of the dwellers E.G First generation - The original members of the fault or people who have come to the vault via the radio system. Second generation are the children of the first generation and so on and so on. This would also help in regards to naming so we know who was the original person with a last name and rename his/her descendants accordingly.