I am having a similar problem, you can totally see the blurriness when running. Since the GC component cables are ridiculously expensive, I was thinking on grabbing a CRT that supports S-Video, would that do the trick? What do you think>
I suppose I can run a test of that.
I don't trust the cables on E-bay. Some of them (especially the ones in the slender box that say SNES S-Video Cable) are incorrectly wired on the inside and basically just split the Composite (Yellow) Cable into two and wire it into the S-Video Cable's end.
The result is: If you use the Yellow, you are using a shitty Composite, and if you use the S-Video, you're getting a shitty S-Video signal. It's lose-lose. You will notice the cable's nonsense when you play because you will see a distinct checkered pattern on the screen. It'll look dithered, as if you were playing the game through a screen mesh window.
So far the only cable that I've been able to get a great S-Video signal is the "Nintendo Video/S-Video/SCART Cable". I'm trying to find an image of it online. I know I took a pic once, but most places online don't show you the packaging (and sometimes they'll swap packaging from what they advertise anyway).
Essentially, this cable (the Video/S-Video/SCART Adapter, though we won't need the SCART part at all) comes with a properly wired Audio/Video/S-Video cable, and a small adapter that turns Composite video into SCART video (for European TVs). You will not need the SCART adapter for this though, as I'm assuming you're in the USA and probably don't have a TV with a SCART input anyway.
SCART is the closest you can get to RGB Video, which is what computer monitors use. But seeing as you'd need an adapter and that adapter is taking in Composite, it won't help. S-Video is the best type of analog video you're going to get if you're unable to afford COMPONENT video (the red/green/blue cable).
Avoid any S-Video cables whose packaging looks like this:
These could be badly wired.Okay, time for some pictures.
Here's the cable I described. It's difficult to find:

Notice that cyclinder under the SNES connection? That tall blue cylinder usually means there's good wiring and shielding in the cable, which is usually a good sign. It's less likely that a cable will have that and not have good wiring of the connectors.
I did a quick comparison of this cable (the AD S Cable Scart Adaptor) and the Gamecube Official Component Cable (which is actually the better choice, but the prices fluctuate from 'holy shit buy it now' to 'holy fuck that's a lotta money!'):
Here's Gamecube Gameboy Player Output through the S-Video Adapter, which is fed into my own RGB Box and into the TV's RGB Input. This is the best way I can get an analog signal that old into my TV. Unfortunately, it also results in a bit of a resolution change, but I'm willing to deal with it.
Anyway, on to the comparison shots.
Here's that cable's COMPOSITE (shitty Yellow) Output:

Pay very close attention (zoom in if you need to, these are really huge pix for a reason) on the Title Screen's Bubbles.
Composite Video shows some really nasty noise trails when the screen moves. Because this is a still screen (only the bubbles move) you can see the noise under the bubbles are they move upwards. This is the worst part about Composite Video and it'll never ever go away. You can minimize it if the connectors are gold-plated, if you re-shield the cables and keep 'em away from other cables, and if you don't use any splitters and just feed directly into the TV... but these options are somewhat unrealistic and won't make the noise go away.
Same title screen, now with the S-Video in play instead of Composite

(you can see the RGB Box on the bottom right)
As you can see, the 'noise' from the Composite Signal is now all but gone. The bubbles do not have any noisy trails under them and everything looks... pretty good. This is the best signal you're gonna get IF you don't buy Component Cables. I'm cheating, however, since the signal is being fed into an RGB/VGA Box. My TV doesn't have native S-Video support.
One thing you will note in the blue background of the title screen... there are these 'waves' of blue. They're horizontal. These waves aren't supposed to be there so this is where S-Video shows its inferiority to Component Video... which is below.
Here's the same screen using the official Nintendo Gamecube Component Cable (again, the best choice):

This is feeding directly to my HDTV's "Component" Connection.
Please ignore the obvious: There's a change in resolution and now things are wider. The Gamecube's Component Cable forces a 16:9 display aspect ratio. Not every older (CRT) Television supports this aspect ratio, so some CRTs won't display this screen, which could be a bit disappointing if you went through all the trouble (and time and money) to get all this cabling only to find out your TV doesn't even support proper Component input (most of what they called DTVs at the time, that time when TVs were transitioning from SDTVs to HDTVs, were not actually supporting all aspect ratios, so you would need to research into the monitor/TV you're gonna get to see whether it properly supports Component input - funny enough, all flatscreen HDTVs these days support Component Video as a legacy type of input).
Notice that there are no waves of blue in the background: The background is a solid blue color. Also, there is almost no noise in the picture, and things are quite clear. This is really the best choice.
You can clearly see that the S-video connection, while waaay better than Composite (which I cannot show you without having to do a lot of rewiring)