Brat, meaning a ill mannered little child is probably from Old English "bratt" meaning a piece of cloth or scrap fallen from a worn or damaged cloak. This word was used in its present sense from the later part of the 16th century. It was earlier used in Ireland for the cloth from which cloaks were made. The implication here is that the bratt was the result of an annoying occurrence (tearing one's clothing) and the little piece, probably ragged and dirty from its wear, was a reminder of that annoyance and not much good for anything.
Another possibility, but not accepted my mainstream sources, puts it from the Old French "brachez" meaning hounds, unruly bitches used in hunting.