In January, Facebook introduced stats per post for Fan Pages. This would be really, really cool, but it's only available to authenticated Fan Pages with more than 10,000 fans. So, "I Bet My Dog Can Get 1,000,000 Fans before Ghandi" will probably be able to check these stats, but many news Fan Pages will be on the outs for a while.
There's some neat goodies, according to Facebook's article discussing the update. With each Fan Page post, there is an additional line of detail showing the number of impressions and Feedback rate (which is based on Likes/Comments, not click throughs or plays).
Potentially more useful, Brightcove released code to add its video players to Facebook Fan Pages. It deals with Facebook's markup language (FBML), but Brightcove makes it pretty easy. The Brightcove page from last September includes a step-by-step guide including adding the FBML app to your Fan Page and provides the player code to make it work.
One minor hurdle is that you need to create and link to a "loading image" (the video player won't load on default). But I was able to create a quick one that gets the job done. My effort (and new video tab) can be seen on the ChicoER Fan Page.
I think it's cool and it's a nice way to bring your video player to your Fan Page (instead of reposting video through Facebook's video service), but I'm not sure how effective it will be. There are other ways to tie Facebook and Brightcove together and Brightcove links to two more of them.