It still wouldn't much live up to it's name.
Advantage or a re-roll is highest-impact when you're close to 50/50 with the check. A typical 9th+ level fighter is looking at having pretty darn good STR & CON saves, between proficiency and the priority the class tends to have with those stats (unless a DEX fighter, of course), so those saves will tend to be excellent - better than 50/50, while INT/WIS/CHA saves are likely to be crap (less than 50/50). Re-rolling the odd failed STR/CON save is probably the best use of 'Indomitable,' giving the lie to the name.
Making it per short rest would reduce STR/CON saves to mere formalities, while leaving the WIS/CHA saves that 'Indomitable' implies resistance to comparatively hopeless.
If I were to run a campaign that might actually go to double-digits, and someone actually stuck it out with a fighter through 9th level, I'd seriously consider house-ruling indomitable to be more meaningful and to convey something closer to actual indomitability. Perhaps restricting the re-roll to STR, WIS, & CHA saves, but also granting proficiency on WIS/CHA saves.