Your kid is always on her phone. She loves watching YouTube videos. You love a bargain. Sounds like the Chromecast, which allows her to blow up those phone videos to TV size, is the perfect back-to-school home video gift.
Your student already has your old smart TV in her bedroom, but she's annoyed that she can't watch all her favorite videos on there because it doesn't have a good YouTube app. The Roku Stick does, along with thousands of other apps.
If your student has a bunch of stuff on iTunes, or she just wants something that looks and feels a lot like her iPhone, the new Apple TV is just the ticket. In addition to numerous TV apps, it also works with many of the games and other apps she might already play on her phone.