Emos aren't depressed. Now bear with me here, because this is coming with background knowledge and etc. etc.
they think that nobody could understand them. Their true selves. Who they are alone at night, thinking and dreaming. This coupled with the fact that they have been shunned by the populars, the 'cool kids', leads to a pretty intense feeling of solidarity. They flock together with the other loan souls, and those that would say they are but are really looking for attention from some funny goth friend
, but this does nothing for their condition. So, "nobody knows me, no-one ever will, why should I live if I have nothing to contribute?"
I find that the best way to help them is to ask them what they think. If they're wrong, suggest a more plausible theory, blah blah blah, and help them to understand themselves, so other people can too. It's a common thing to blame the music, but that kind of suicidal emotion is deep, and no amount of teenage phases or rebellions can cause it. So, let your little sister talk to you about stuff. Not peers, but meaningful stuff, like life, and reassure them that there is boundless potential in the form of creativity or logic. You might just help make a little genious.