I thought it refered to the size of her....never mind, but I later found out that it's short for Diamond dandruff (I forget which episode it said this, it was the only one I had ever watched really). In kindergarden I think it was, she was shocked by the class pets (Plusie and minun) which caused her hair to get all...sparkly. Her classmates instanly made fun of her and gave her the nickname diamond dandruff, or DD, a name she still hates to this day.
Kenny and Leaona call her Dee Dee because when they were little, Dawn's hair sparkled because a Plusle and a Minun shocked her. The "Dee Dee" stands for Diamond Dandruff or something and everyone made fun of her for it, etc.