Your "SYS" light seems to be voltage related; a new battery "cured" the SYS light for a time, and dash backlighting brightens and dims...intermittently, yes?
I'd want to see what the voltage across the battery was with the engine running (charging), and to see what it was when the dash backlight dimmed (and SYS light came on); i.e., is the alternator intermittently dropping the load?
If the battery voltage is between ≈13.8 to 14.7 (give or take a few tenths),
and remains steady whether the dash backlighting and/or the SYS is lit or not, coupled with NO drop in rpm or other indication of increased engine loading, a connector or ground connection would come into focus. Refer to your FSM (of course we ALL have FSMs, right???).
However, if battery voltage drops (to 12.8 or less) in coincidence with the dimming/SYS problem, then the alternator or connection to it (e.g., field connection, etc) becomes suspect.
Anyway...A leeeeetle bit more info/results from some measurements before and after the symptoms occur would help narrow the field of possibilities...
P.