With multiple sources to a TV, and no HDMI switch available, I've run component. Where HDMI is available: I mostly run HDMI (my cable boxes are all hooked up component because, bluntly, the box puts out a better picture through the component outs).
This is the reality of the world... Component video is capable of carrying signals of up to 1080p without issue, but most A/V gear doesn't output video above 1080i over component video.
So, which is better HDMI or component if all you are dealing with is 1080i?
IT DEPENDS!
Yep, that rock solid answer which doesn't help you at all.
Because most video has a fair bit of processing involved with it, you get mixed results from device to device and from TV to TV as to how well it handles component video vs. HDMI video.
A game system could output great 1080i over HDMI, while your cable box may output better video over component.
The resolution, despite being the same, still needs to run through digital to analog coverters, scalers, and more before being output to your TV.
So, you really have to try both component video and HDMI to see which truly looks better.
Don't mess yourself up by spending cash on some overpriced cables though - go hit up
www.monoprice.com and pick up plenty of decent cables for fifty bucks or so and give it all a try, then decide for yourself which version is best. It really is the ONLY way to know for sure.
In my experience:
HDMI looks better than component about 45% of the time (marginally)
They look the same about 45% of the time.
Component video looks better about 10% of the time. (marginally)