My W. actually told me she would get the presents from both of us this year, and put both our names on them, or simply "Santa". That's OK with me.
That makes it easier. Sounds like a thoughtful gesture. However, the kids will know she bought them and put your name on them. I am sure that is ok with them considering you are so far away. A surprise from Iraq would still be cool I think! Is there anything you could buy that would be unique? I have no idea about the shopping situation there. Anything arabic? Anything with arabic writing on the label? I was thinking of something like the men's head covering (at least the ones from Saudi Arabia make a great little tablecloth). A tiny arabic coffee pot or little vase? A set of camels? Just a small token gift....even just a stocking stuffer....just a special touch just from you.
I am not saying this to be critical.....just a different perspective.