I am a music major, so a large part of my application was the audition. But my friends and I have talked about admissions, etc., and this is what I can tell you:
Have the interview, and be yourself! It's a lot of fun.
Do well in school and on standardized tests (well that's a given).
It helps to have found something you're passionate about, of course, to list under extracurriculars.
Recommendations: from someone who knows you well.
I'm convinced they see how NICE students are, because it seems that Rice students are nicer and friendlier than at a lot of other schools. I noticed it when I was visiting colleges, and I still notice it. But I might be wrong; that's just a guess.
A good fit: they want someone who will fit into this university well.
The essay prompt was difficult for me, so I just ended up writing in detail about one of the most inspiring and passionate "aha" moments of my life. I guess it didn't really address the prompt they gave us, but I loved the essay and felt it represented me well, so I sent it in. Hey, it worked for me.
Don't ignore The Box on the application. Most people I know put a lot of effort into it, and came up with very creative things.
GOOD LUCK! It's a great school.