Here we go:
Family Tree - When viewing a dweller you may look at an extended family tree tab. This way you can scroll through and see who is related to who. Just a neat little feature to make it feel more immersive.
Taking Mass Inventory - An option to view all equipment and weapons currently on all dwellers. I forget sometimes who is holding that one awesome gun and sifting through all the dwellers is a bit tedious.
Emergence Crew Room - Dwellers assigned to this room will act upon any emergencies. They will run to the room in need and aid the situation without replacing the dwellers currently in said room.
Exploring Decisions - "Your dweller finds a ruined Super Duper Mart. Should they enter?" Stuff like that. Some decision making would be neat. There would be a timer on it so that your explorer can continue if you are not looking.
Day & Night - Outside the vault you see the time change based on your devices time. Adds a little more immersion I think.
Robot Assistants Bay - Mr Handy and Protectatrons would be awesome to have. The Robotics Bay is a room that gives you a robot of your choice. (You could have 3 on a 3 merged room and up to 6 fully upgraded.) Robots must return to their room for repairs after incidents.
One Mr Handy robot can be assigned per room to assist in resource production and are very good at killing roaches and stopping fires.
Protectatrons are very good at stopping raiders and can be assigned to the vault door.
Day / Night - With a day and night cycle dwellers may take shifts to operate a room allowing idle dwellers to contribute when there is to many people and not enough rooms.
Stay tuned if you like. I'm sure I'll come up with some more stuff as I play and stuff. Till then.