3 main approaches, in addition to painting it/wrapping it of course.
1) Abrasive approaches. Here you will use abrasives of various roughnesses (grits if you prefer) to get rid of the scratches and take it back to a shine. Very annoying to do properly on plastics, much less on curved surfaces, and as you are removing material then you are also having that problem, even more so when the dyes maybe did not get that deep.
2) Chemical/solvent "wipes". Don't know what plastic the PS3 shell uses offhand and there are some solvents that will mess up otherwise very strong plastics (see polycarbonate and acetone, and a blend of polycarbonate and ABS or something will often be used in things aiming for a bit of strength).
3) Fire, or flame polishing if you want to be professional (if you are not at least thinking of doing your best
Arthur Brown impression I don't think we can be friends). Usually used more for crazed and faded goods, and acrylics, but not exclusive to them
For 1 I would note toothpaste as a better potential household good (toothpaste being basically sandpaper in gel/liquid form for teeth) but might still be too harsh, though some of the things aimed at automotive headlights might be an idea. Some will also investigate things aimed at metals - brasso does more than brass but its use with plastics, much less glossy ones, I am less familiar with.
Whether you want to stick to hand techniques or find yourself playing with a bench grinder, rotary tool, drill or something like that I don't know -- power tools can be more consistent and certainly quicker but there is a learning curve rather stronger (not to mention potential to screw things up).