The problems I see are:
1.) Why get me, some random person to pick a city of my choice, if they have that type of ability they could pick any targets of value (Washington, New York, London etc...) with relative ease and avoid me being a smart ass and going for somewhere like Chernobyl.
2.) If I decide to die along with my family, what's stopping them from dropping the nuke regardless, or repeating the process with another person. - NOTE: I just saw the OP's edit, but I want to keep this in here.
3.) They probably won't allow you to choose an abandoned city.
So, cold, harsh logic dictates that I should choose a city to avoid something worse from happening.
So, let's say they'd allow me to choose a city as long as it has a population over a certain amount. The chaotic side of me says to go for a major city in world trade and see what the repercussions are. I'm also slightly curious to see how the world would react to this and what would happen to the country in question.
However, the most ideal path (in a matter of speaking) would be to target a place in the developing world which has a very high crime rate, homelessness and poverty in general. As this could attract attention to the rest of that nation and the world would bring along improvements to that nation somewhat.
That's my decision, I wouldn't want to kill my family as I have, obviously, deep bonds with them. While on the other hand I'd probably (for a while at least) feel rather disconnected when people I don't know would die.