I'd get a new one. You can only clean them to a point IMHO and the green/rust seems to stick no matter what. I buy wooden ones for about £20, they only take me ten minutes to fit. The last time I kept trying to clean the green, it rusted the screw and was a nightmare to change in the end - just warning you not to let it get that far!
Omg, I thought it was just me who had manky hinges! We replaced all our loo seats when we moved in (plastic for wooden John Lewis) and they soon went green. Nothing shifts it but I might well try Barkeepers friend.