I like to watch Youtube feeds for SpaceX rocket launches. Being able to watch both the SpaceX official channel and a commentary channel like EveryDay Astronaut - and being able to flip between them - would be great. That's just my main use case, of course, and I can imagine lots of others.
Conceptually, this wouldn't rely on any sort of capability within the TV/monitor you're using. This would instead need extra hardware and software within the Roku box itself to be able to stream two distinct feeds and then manage compositing them into the HDMI output to the screen. Add a dedicated PIP button into the remote and Roku would have an absolutely killer product that (as far as I know) nobody else has.
I bet it would get some great press and for me it would be easily worth the premium cost it would take to put this on my main TV. This would be a HUGE feature addition for Roku. Roku Ultra Ultra? Roku Ultra x2? Oh, please make it happen...