On the gaming SE site YouTube videos are automatically embedded into answers. Could we get something like that too?
It would be very useful for showing mechanical parts/creations in action, especially LEGO robotics.
The best way to make a case for this is to demonstrate a need for video on this site. If video would improve your answer, start linking to videos in your posts!
Assuming they're actually useful, this provides some concrete data for judging how and where they're appropriate, lets you create guidelines for their use, and lets us see that adding support for embedding them directly would be beneficial to the site.
Questions based on robotics and mechanics tend to include videos, which can help explain a complex mechanism. A video alone isn't usually enough to answer a question so users should be encouraged to expand on important points.