The weekly quest "Game Show Gauntlet" will give you 4 dialog options. I haven't seen it yet myself but that is a good one to go for. Here is also a link to the answers so you don't have to fight anybody. I didn't make this list and all the credit goes to the person how made it.
http://ingameinfo.blogspot.nl/2016/09/Fallout-Shelter-Gauntl...EDIT: 22/2/17
So
Yin found out that you can answer the first
three questions in the "Game Show Gauntlet" and then "Give Up" the quest in the objective screen. Then you retry the quest, witch will cost you some caps. Starting at 200 and then increasing every time you retry it. The exact amount it goes up I don't know. But this makes the achievement even easier if you can spend the Caps. He estimates that you need to save up about 20000 to retry the mission the 8 times to get the achievement.
Thank you Yin for sharing.
If you can't spend the caps these are also good options to go for. You don't need to do it in one quest it's cumulative so don't worry about it if you can't do it all at once.
Some of the quests have 0-1 dialoge options. But from my experience there are more quest that don't have options than there are quests that have. Until I got to high lvl quests.
I also found that if you have dwellers in the wastelands and they come across a location that you can Interact with. (a map with a audio cue will appear underneath your dweller counter. Picture below). This is completely random but when you need to clear out raiders there was always a dialog option, these also count towards
Legend of the Wastes in Fallout Shelter
Also with some quests later on where your dwellers need to be lvl 38+ I had a lot of quests where you would get 2-3 dialog option. By then I already had the achievement. But might be useful to know.
EDIT: 18/2/17 fixed some grammar
EDIT: 19/2/17 Added a picture of the icon.
EDIT: 25/2/17 Added Yin's findings and reformated the solution.