I also can give recommendation to Novus.
I tried numerous plastic polishers (basically equivelent to Novus 1) but then saw lots of posts about how well Novus worked.
I bought some on a great deal off of eBay (1/2 what others sold them for---be careful, some only include trial packets of Novus 3, you want the entire 8 ounce bottles of Novus 1+2+3).
I tried it and it did wonders for restoring old Nintendo push buttons (a requirement since they are about 1/16" narrower than all other pushbuttons and shaped differently and no longer made), plus they fixed a portable Coleco Frogger's screen that someone mistakenly cleaned with orange cleaner and softened the plastic making it totally opaque

It brought it back to near perfectly clear screen.
I can't imagine working refurbishing old arcade games without Novus. It is great stuff!
(And no, I am not a paid sponsor, just a very happy user of their product.) It won't do miracles like 1/16" gouges or something, but surface scratches and opaqueness and dullness it will definitely fix given enough elbow grease or low torque power tools and polishing pads.