So, to get this started, I want to throw out a theory I've been developing based on some pretty blatant hint-dropping the show is doing that suggests Katagiri and Hinazuki are related—probably sisters. Their eyes and hair color match perfectly (see the red puffiness around them), and their voices actually sound quite similar (to me, anyway). If they are sisters, then Katagiri would have had to have been not yet born in 1988 (assuming she is ~17 in 2006). It is also explained at one point that Hinazuki lives alone with her mother, which means that Katagiri could not have been born yet.
So, let's say they are sisters, and that Katagiri has not been born yet. It is also unusual, to say the least, for someone to have kids 10+ years apart. My theory is that Hinazuki's death prompts her mother to have another child—Katagiri—and that Fujinuma will be forced to choose between saving Hinazuki, which could prevent Katagiri from ever being born, or not doing so knowing that she will be. This is the kind of thing I would expect out of a normal time travel show, but Bokumachi has already shown us that it is anything but normal, so I'm not that confident in this theory myself, but what do you all think?