What exactly does the MB-326's 'relight' button do?
The Aermacchi MB-339 was developed from the -326 and it is also powered by one Rolls-Royce Viper Mk. 632 turbojet. From its aircraft manual, the relight button is for the in-flight re-ignition:


  Premuto - Attiva il sistema di riaccensione in volo.
  Pressed - Activates the re-ignition system in flight.


The start button in your picture is for activating the starter, and the guarded ignition next to it is likely a ground/start-only ignition mode.

Typically, an in-flight ignition mode is of a lower energy rating compared to the start mode, this helps prolong the life of the igniters. It's used when flying in rain or icing conditions, and for maneuvers that could momentarily choke the engine (e.g., an extreme spin or side-slip with side-mounted intakes).