This may sound a bit blasé, but a well-faked demo of a totally original 2040 technology is going to be more well received than a well-designed product with 2011 technologies.
This varies depending on the particular group at the lab, of course--some groups, such as Biomechatronics (http://biomech.media.mit.edu/) are more grounded in reality, whereas others, such as Lifelong Kindergarten (http://llk.media.mit.edu/) are more experimental.
My advice: READ THE PAPERS that Media Lab groups are publishing. Especially recent Phd theses, and papers at the high-profile conferences (SIGGRAPH, SIGCHI). Think about them. Write about them. Figure out how you'd build upon them. Then talk to the authors of the ones that excite you the most.
NOTE: This is based upon my own experience, and my views do not represent those of the Media Lab. I've worked at the Media Lab in the past, but I'm not speaking from any official capacity here.
I don't think knowing how to show off is necessarily more important than being a good artist/innovator/engineer, but it's at least as important. The Media Lab wants to show off too, after all, and they need people who are good at it.
I'm not sure how you see the Media Lab as different than other art/design schools in that respect. It's definitely different than engineering schools, yes, but it's not an engineering school.
According to him, word of mouth is paramount. Who do you know there?
As he put it...treat this as a the interview. If you can't befriend someone who works there they you aren't qualified.
Prof. Raskar has a good set of advice on the Camera Culture page http://cameraculture.media.mit.edu/join
I'd recommend reading all of the pages relevant to the groups you want to be a part of. I second Storborg's advice to read some recent graduate papers.
I'd also recommend working on your site specifically for the eyes of MIT professors. Highlight your work and how it applies to the world outside the field in which it was made. Talk about what inspires you and what you want to pursue in your own work. Think of it as an ad campaign with your professors as your target audience. You need to sell them on why you are right for that lab. Ask yourself the question "why should I be at Media Lab versus anyone else" and then answer it through your portfolio.
Eyebeam is amazing. Or, that is to say, the group I worked with was an amazing group of people. There were lots of creative people making the shops hum, teaching classes, and sharing ideas. Though, the group of people using it as a work/research facility changes almost entirely every six months. It's possible that the personality of the group changes substantially with it.
It's universally guaranteed to be brilliant. There are also a lot of dramatic transformations the whole place goes through as different people change the gallery spaces to suit their projects. While I was there working on Fairytale Fashion it changed from a formal art gallery, to a small faux bookstore, to a museum, to a fashion runway. It was sometimes like watching mushrooms grow in fast motion.