Will Strong, Black Coffee Help Cure A Hangover?

In a  word NO.  In fact strong black coffee will make your hangover worse.  The hangover is caused by dehydration.  Alcohol makes your body urinate more, since alcohol is a diuretic.  So you wake up feeling tired, with a headache and often a little bit shaky.  Black coffee, particularly if it is strong will also cause you to urinate, since caffeine is a diuretic.  So you will lose even more body fluids and your hangover will be worse.
In addition , through increased urination, you have probably lost blood sugars which is why you may feel shaky.  The caffeine may cause your heart rate to increase, which will make any 'shaky' feeling you have decidedly worse.  So coffee is not a good idea.
The best thing you can do is to drink lots of water and try to have a good breakfast with lots of carbohydrates to make sure that your blood sugar levels are eased back to normal.