TL;DR - Not much you can do, call a luthier to find out about electroplating or buy a new tremolo. You can try metal polish but it will likely remove some gold finish (I wouldn't do it)
From my experience there is no easy solution for this. Corrosion is corrosion and as such it's metal that has been converted and destroyed. You can do some cosmetic cleaning but really the unit needs to be taken back to bare metal and electroplated. The early Cosmo black edge tremolos were notorious for corrosion and in the early days the gold hardware from some asian manufacturers had some serious issues. Everyone's sweat is different and some can be more corrosive than others. No matter how much I clean my late 80's JEMs they continue to corrode (ever so slightly). Obviously replating a heavily damaged original '88 floral edge tremolo is worth it, in order to keep it as close to original as possible, whereas a less collectable guitar it's easier to just replace the trem.
I guess it comes down to how much it bothers you and what the guitar is worth to you to sort out. Most luthiers have an electroplater that they use (if they don't do it themselves in-house). I recently had a quote for $100 USD to get my '89 VBK edge replated to match the original Cosmo black electroplating. A replacement trem was $260 USD from Rich at Ibanez Rules so I'm using that unit temporarily until I get the original back from the luthier, I will then have a spare 'new' edge trem to use for parts.
I pull my tremolos apart every 18-24 months and scrub everything with a toothbrush (that I clean with lemon oil when it gets dirty). I'm not recommending the lemon oil part, it's just something I have done for 20 years to clean off the build up on the toothbrush, I'm sure someone has better options (oils attract dirt and dust which you want to avoid). I then use a dry clean cloth to wipe down and polish the parts before reassembling.
Make sure you wipe down the guitar, strings and tremolo whenever you play as this will help prevent the corrosion from starting. I have noticed that electroplating in general has improved significantly in the last 10-15 years.
Not sure if this helps, but hey it's free advice and should be taken as such