After thinking about it a bit I am inclined to think that if the theory of abuse holds water I think it is much more likely to be between Menma's father and mother as opposed to between Menma and her father - for example if Menma saw her father physically abusing here mother. Of course this could lead to all kinds of variations as regards "why", but I also agree with Solace - it is true that my speculation IS just shooting in the dark for now. And I am notoriously bad at guessing where anime/manga plots are headed!
But for now I am still going to go with my theory in general and put the focus squarely on some sort of abusive activity between Menma's mother and father that she unknowingly stumbled across.