Somewhere it was suggested that your game's saving throws should be named for general things that might happen to players over the course of your game. Someone else once made a post linking each of the saving throws to a demon prince in the Fiend Folio. So a character making a saving throw was like thwarting an evil force-- pretty cool.
I think the most difficult part to the saving throw is just how they feel like an arbitrarily high d20 number and there is not a lot of immediately comprehendible difference between classes. So what if instead of numbers we used increasing die size (a trick Perdition uses for skills) and attempting to roll a 6+?
With any B/X retroclone, a first level Fighter might start out with all d6s. A dwarf might start all d6s, but a d8 for Poison. A halfling might have a d8 for Breath. Elf and wizards might have d8 for Magic. A cleric might start with two d8s based on their god.
What about the Lost Serpent King & The Broken Forest of Ynn? My sorta Sidney Sime-inspired fairy tale romp?
Taking the rule that saving throws should reflect the adventure, maybe the saves (save on a 6+) would be: Confusion, Illusion, Eternal Slumber, Imprisonment, Transmogrification, Fortune. The idea is that death by damage is avoided, but protagonists are often put to sleep, imprisoned, cursed, transformed, lost and sometimes eaten (a type of transmogrification; swallowed is Imprisonment).
Humans and Halflings have higher Fortune being lucky (d8). Dwarfs, with hearts of stone, might suffer to Eternal Slumber and Imprisonment (d6) but strong against Transmogrification/Confusion (d10). Elves being of magical origin might do well against Impresionment/Eternal Slumber (d8) but easy to anger and distract (so weak against Confusion and Illusion; d6).
UPDATE: To circle back to the first point about save throws reflecting the world. Would it change the feel of a game if the saves where the seven deadly sins- Pride, Greed, Lust, Envy, Gluttony, Wrath, and Sloth? Better to use the 7 virtues? Are various hexcrawls and modules improved by changing the saving throws? Would Carcosa be better with differently named saves? What would Hot Springs Island be? Fire, Water, Blood, Sugar, Sex, Magik?