I had the same problem. I hooked up the rca to see the screen with rca selected on the xbox cable and maybe under settings too in a menu. Then back to componet and turn the xbox on and off with componet. Worked ever since. I have an old xbox 360 with no hdmi so not sure if the new ones have the selector on the connection when it goes into the xbox, but basically after switchin back and forth the componet worked and no problems since.
It may or may not be helpful to you. I was successful in using the component video with the Wii. See if you can use the component video input to the tv using a different source other than the XBox 360 if you have not tried that already. If another source works, then it's definitely a setting in the Xbox as j.Howser has suggested.