I've been searching for a direct quote where Nolan addresses the reasons why this song was included in the script. All I found was this transcript from a press conference which included the following exchange:
Hollywood News: The score and the sound design for this film are phenomenal; it’s almost like another character. Chris, can you talk about that a little bit, how you constructed that?
Nolan: I like films where the music and the sound design, at times, are almost indistinguishable. And one of the interesting things that happened early on is the Edith Piaf song [“Non, Je Ne Regrette Rien”] was always in the script — long before Marion came on the film — it had always been that choice of song — and right at the beginning of our post-production process, I had to make the decision of “Do I get the sound department or do I get the music department?” Do I get Hans to manipulate that track until it sounds as if you’re hearing it through the dream, where it slows down and gets massive and all the rest. There was an interesting way to go; what I decided to do was give it to Hans and let him run with it and see if in some way it might inform elements of the score, because we always knew, we talked in early conversations about how towards the action climax of the film, there was going to be a need for the score to interweave seamlessly with this source cue, which is and extremely difficult technical thing to do.
That quote doesn't explain why that particular song was chosen, but it does explain that the song was chosen early in the writing process.
However, I don't think it's necessary to get a straight answer from Nolan to this question. It's pretty evident from the lyrics that it has a strong connection to the theme and plot of Inception.
Here's the lyrics translated to English:
I regret nothing
no, I have no regrets
I regret neither the good things that were done to me nor the bad things
They are all the same to me
The past is payed, swept away, forgotten
I don’t care of the past anymore
I set my memories on fire
My agonies, and my pleasures
I don’t need them any more
Swept away in the agonies of love
Swept away forever, I’m restarting with nothing…