1. Make a cool video
2. Post to reddit
3. Vote it up
Also, make a cool video or take some good photos and get it featured on Wired, Gizmodo (as well as the other Gawker sites with a lot of traffic), Huffington Post, local newspapers, etc. Telling a story of triumph and providing hope for the future will get the attention of readers and be shared throughout the Internet.
Viral is about understanding what strikes people in a way that they have to share. For instance, everyone loves cat videos. In fact, it's at least 64.3% of the Internet. Everyone loves videos of cats chasing laser beams.
I propose a modest adjustment to this approach: make a video of a dog catching frisbees from an FRC robot. However, don't just make it a robot shooting frisbees, make it a robot sidekick with an action hero dog (with a cape!). Perhaps it could fight criminals offending the
Order of Operations or illegal
Trigonometric Identities.

I think that's good for at least 2 million views.
In all seriousness, I think getting a quality video that tells a story in under 2 minutes is key. One thing that connects everyone on a basic primordial level is the childhood dream. Developing a video that shows how FIRST has helped students achieve their dreams will help foster a working platform to capture an audience. Perhaps connecting with some successful FIRST alumni and detailing their story will help establish an outlet for mentorship and guidance.