I have this issue. I had it before ICS and I have it after ICS. It is intermittent, so it doesn't happen all the time and it happens with people texting me from all types of phones. I don't know what it is and I don't use the stock SMS app. I use Handcent, which I thought was the culprit but hearing that some of you are getting this using other SMS apps, it seems like maybe the phone or network related.
It's weird. The second duplicate text would come minutes after the first one so I get fooled into thinking it's a reply.
But it's so intermittent that I can't predict when it would happen next.