Fame is increased by killing, by dying, and by gaining experience.
Have enough fame to be on the Hall of Fame, and not be a pretender (and also not be any of the unique summons, yes?), and you get a heroic ability that increases so long as you're on the Hall of Fame.
The ability might not be all that useful (ex. Heroic Stupidity might be bad on a mage, and Adept Researcher may not be what you had in mind for a heroic, bloodthirsty Niefel Jarl), but they can be quite nice (extra hp, extra quickness, extra reinvigoration for mages, etc).
Since dying increases fame, if you have nothing better to do with an immortal on a turn, it has no extra gear to lose, and you don't mind it being resent to your capital or possibly bumping somebody off the HoF, it might as well die.
Death ~ Fame
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