I'm using component right now. I was thinking maybe I should just buy a new arcade and then transfer my 60 G hard drive. I would get the built in hdmi and I believe a better chip set/processor so it runs quieter. Plus since this xbox is about 6 years old it would renew my warranties.
It still doesn't matter.
Well, sort of: I'll elaborate.
There are two important resolutions here, the render resolution and the output resolution. The 360 is, was, and always will be utterly incapable of rendering all but the simplest scene at 1080p - so let's call the render resolution 720p.
The output resolution you're on now is almost certainly 720p - that's being sent to your TV, which then upscales the image to whatever your TV can output.
If your xbox was capable of outputting 1080p (I don't think the original ones are? Could be wrong on that point, though) then it'd *still* render at 720p, except it would upscale before it sent it to your TV, which would simply display it without upscaling it.
Now, if you have a half decent TV, there is absolutely no advantage to your 360 outputting in 1080p. Cheap TVs can have very poor hardware upscalers, in which case there is a small advantage, as a good upscaler can produce marginally better looking images. But this is certainly nothing worth spending money on, because whatever way you do it you're not going to get *any* more data in that image.