What causes Black or Brown water the morning after system washes it's self?

* Black water can be oxidized hydrogen sulfide, or in severe cases, from Iron or Manganese.
OR
if you have CARBON in the tank ( or in any tank installed ahead of water softener ), it could be the carbon breaking down ( New Carbon required to solve that black water issue ).

If NO Carbon is being used in any tank, then it's just a build up of oxidized sediments that might go away with a few extra regenerations or longer Backwash time ( put into Backwash, and unplug timer for 1 hour and let it flush ).
If none of that works, then it's time to RE-BED the Tank ( dump out the media and re-fill with new ).
If water softener, see:
Resin Replacement, When, Why, and How

If greensand or other media tank that is the "Source", then you'd follow these directions,
SEE: How To Replace Media ( Carbon, Birm, Greensand, or Calcite ) in Filter Tank

and if Carbon tank is used,
see:
Carbon Replacement Guide

Or if you know that CARBON was Added ( included ) in the water softener resin tank,
then see this article:
Why Not to Mix Carbon with Resin

* The small amount that some manufacturers add ( Rainsoft and Ionics are two, but there are others ) does next to nothing after 3 months.

If you want to remove the chlorine in a water source, you need to use a separate carbon filter ahead of the softener.
see:
Whole House Carbon Tank

Or on Well Water, you'd use one with a Backwashing control valve,
see:
Carbon Filter 1 cu.ft. with Fleck 5600 Valve Backwash Only