This is the type of valve or whatever on our kitchen faucet; from time to time over past years it develops an annoying occasional drip after shutting of the faucet and to stop it I end up having to replace that black washer thing as it apparently wears down after use. Usually takes care of it but this time I still get the drippage, improved but not stopped after replacing the black washer thing. What else could be causing the drippage as described, after this "fix". It's a pretty old el-cheapo brand faucet does that have anything to do with it maybe? Any comments appreciated.

That kind of seal originated with Delta, others have copied it. So your faucet may be Delta or a knock off.

Are you lubricating the black seal? You should. I hope you are replacing the spring when you replace the black seal.
Perhaps the stainless steel plate at the bottom of the stem is worn, in which case replace the stem. It looks like a Delta stem.

Lastly the black seal has to seal against the wall of the hole in which it resides. Make sure the hole is nice and clean and smooth.

sometimes you take the spring in your fingernails and stretch it a little. this gives it more more push against the rubber.