I'm sorry, but I always thought Aeris was annoying. I was never at all attatched to her, and when she died, my reaction was something along the lines of "sweet."
Now, in FF6, when Celes tried to commit suicide after Cid's death, I definitely cried. That was a brilliantly executed scene (ignoring the obvious soften-the-blow ploy with the Cid quote right before she jumps), at just the right point of the game, and it really drove home that feeling of dispair that was so appropriate for that part of the game. And of course, the fact that Celes was a great character all along (and had a heart-breakingly beautiful themesong) made it that much more powerful.
That's about the only time I can think of where a videogame has left me in tears.