Re: Stream Google Drive Videos to TV
The Chromecast can stream video from places like YouTube and Netflix, and can also stream a Chrome tab. So, you could drag the video into Chrome and then cast that tab to the Chromecast over wifi. It's a little laggy but works OK.
Also, since it's Google, I wouldn't be surprised if they offered options to stream content from Drive to the Chromecast in the future (although you can't do it now).
Another, probably better, option would be to get an HDMI cable to connect her phone to the TV. Then just plug the phone into the TV and play the video. You will need to get an MHL cable for the S3, which is $10 direct from Samsung (or cheaper from other places).