I just tried that myself last week, although mine was chicken rice this time. My chicken rice soup is pretty much identical to my chicken noodle.
I simmered the chicken I had left over from a previous dinner for 2-3 hours to make the stock. This chicken had been cooked in my PBC with a little apple wood for smoke, and had been seasoned with Plowboys yardbird.
After straining out the chicken I throw in chopped carrots, onions, celery and some chopped parsley. I let all that simmer while I pick through the chicken. I let that simmer for about an hour, threw in the rice and the chicken, simmered for another 20-30 minutes til the rice was cooked. Adjusted the salt and pepper to taste, some times I'll add a little chicken base, just depends on how "chickeny" my broth comes out.
It turned out real good, had a mild background smoke flavor, not overbearing. And the Yardbird added a nice flavor profile as well.
The one thing I do differently with chicken noodle soup:
If I expect it all to be eaten that night, I cook the noodles right in the soup just before serving.
If I am making it to freeze and or hold for later, I cook the noodles seperately and add them to the soup after it has cooled considerably, as I don't like my noodles to turn to mush, just my preference.
This freezes well, and reheats great, and the noodles still have a nice firmness to them.
Bottom line is I liked the little bit of smokey flavor that I got, I don't think I would have liked it with a more powerful smoke tho.
Good luck, give it a try and report back with your results.
Weber Performer with rotisserie ring, 22.5OTS, Maverick 732, Weber Smokey Joe,PBC Smokenator/Hoovergrill, variety of microbrews, Gin and tonic, variety of lighters.