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.
I haven't played the game up to 9th level yet, so I haven't personally see Indomitable in use, but I must say that so far I haven't really understood it.
First of all, from the reading of RAW, Indomitable is
not equivalent to advantage. It doesn't grant you advantage on a ST, it lets you
reroll a failed ST.
Maybe I am reading it wrong, but to me it's not the same, and not just because you only use it
afterwards, but also because IMHO this RAW means you reroll the whole ST
including the second dice if you had (dis)advantage.
But this still doesn't tell me how good it is. If you had advantage and failed the ST anyway, with Indomitable your chances of success are greatly increased. If you didn't have advantage, Indomitable is practically the same as advantage (except if truly having advantage would kick off some other ability of yours). If you had disadvantage, Indomitable is probably worse than advantage. Maybe we can assume that most of the times the Fighter doesn't have (dis)advantage and so Indomitable is practically identical to advantage anyway.
Also, during the playtest before the release of the game, Indomitable instead really granted advantage, but
all the time, not once or thrice per day!
Not sure, maybe the designers considered how many ST on average a PC is subject to, during a typical adventuring day. Maybe they considered 6 encounters/day, not all of which feature monsters with ST-based abilities (some of them might force many ST in an encounter, but not all of them will target the Fighter), and finally the Fighter will use Indomitable only after failing the ST. So in a sense if e.g. we have to roll 6 ST in a day (an average of 1/encounter) and maybe we already save half of the times, then Indomitable could negate 30-40% of the failures. It would still be a decent benefit, but all this is based on too many assumptions.