You can do it, but it's not terribly cheap. You need something like this:
The plug in the first answer won't get you there; it's for specific video cards that are capable of putting out component analog signals on particular pins on the DVI connector. You can't turn component video into DVI or HDMI with a passive adapter, since you have to convert the image from analog to digital.
Edit: if your monitor takes only digital and won't take analog VGA on some of the DVI pins, I don't think the Wii -> VGA -> DVI adapter route is going to help you either. There are VGA -> DVI converter boxes like the one above that take in analog VGA and put out digital DVI, but the ones I know of are super-expensive since they can handle much higher resolutions than your Wii's going to put out.