I think I'd have to have some kind of tie to the country in question as opposed to "country shopping."
A buddy of mine got to run in the World Championships this summer because he represented his birth country. He's been in the States since he was pretty young and loves it here, but he's got plenty of family back in his birth country and feels strong emotional ties there as well. I felt like his situation is a beautiful thing- high level racing opportunities aside, he gets to embrace America and remember his origins.
For me, on the other hand, my family has been in America since the potato famine. I don't really have any ties to other countries and would feel weird about representing some other nation in international competition, as cool as getting to run in the Olympics would be.
On the other hand, if I had moved to another country for whatever reason,(job, wife, just 'cuz)lived there for a while, and could see myself being a long-term resident of that country, I would feel good about representing them internationally if they would have me.
I wouldn't knock anyone who did country shop to go to the Olympics, though. I can totally see the appeal, it's just not for me without some extenuating circumstances.
Here's a followup question:
If you could represent any other country EXCEPT your home country, what would it be (this assumes the country in question would absolutely, 100% name you to their team)? I'm thinking either Ireland or New Zealand. I go with Ireland because while my family's emigration from there happened well over 100 years ago, it's still the country from where my family originated and that would be cool. My second choice is New Zealand because 1) the running tradition they have there 2) a dear Kiwi friend of mine once told me I'd make a good Kiwi myself and 3) their uniforms are BOSS.