The iPod mini is brushed, while the iPod 3G/4G/photo/etc are not... the term you're looking for is simply "polished" or "mirror-finish." Brushing dullens (and actually protects, scratch-wise) the metal look.
it would be hard to brush the mini as the natural finish of andonized aluminum is sort of brushed. Also bear in mind that the color does not exist all the way through the metal casing and you risk stripping the color off the mini if you're not very careful.
You can give it a brushed metal look by soaking the housing in Easy Off over cleaner for about 20-30 mins (to get rid of color), then washing off the residue (remnants of anodize finish), then follow the same procedure as the regular iPod brushed tutorial.. or use a fine grit sand paper.