The study of weed origins and evolutionary history is the poor cousin of the archaeobotany of crop domestication. Archaeobotanists can potentially do much more on this, and undoubtedly should. To provide some inspiration it is worth considering some recent insights from genetics, to do with weedy rice. While it is surely the case that rice's wild progenitors may act as weeds in the crop, it now appears that much weedy rice is descended from the crop and not directly from the wild progenitor.
Via Dorian Q Fuller, Eve Emshwiller, Jean-Pierre Zryd