I have a black colored copper sink and thanks to lye that I poured to fix a slow drain, there has been a discoloration .
How do I paint it back again?
Your sink probably isn't painted. Copper goes through several stages of oxidation (roughly the equivalent of iron rust). The final patina is the dark, durable finish you see there. Painting it would've completely defeated the point of building it with copper in the first place.
Over time copper will naturally change colors – transforming from a shiny brown color to darker browns, then blues and finally greens after a number of years. When exposed to the natural elements such as wind and rain, copper develops this “patina” which actually protects and preserves the metal underneath.
Your sink probably doesn't ever get to the green that article mentions due to the regular wear of use and cleaning. You'll want to just give it time to re-oxidize. There may be chemical methods of accelerating the process, but you risk doing more harm than good.
Try some natural stuff first. I/E: Citric acid (lemons, limes or oranges), vinegar and/or baking soda, or peroxide.
I play drums and I use Brasso to clean the cymbals. Brasso is some powerful stuff. I know that if you leave it on the cymbals too long, it will eat the logo and/or any type of finish that some cymbals have. Cymbals are composed of tin, copper and silver but your finish is a black oxide. As with any chemical, try it in a small inconspicuous place first (like only over that stained area) before applying it liberally.