DEFENSIVE
Resistance. Make save with +5/+10 modifier.
I like a lot of these ideas, but this one above gave me an idea. This is just off-the-cuff spitballing at this point, so there may be things I have not thought through fully yet.
Instead of Legendary Resistance, legendary foes get a certain number of
Legendary Destinies (LD) per round.
Legendary Destiny: Once per turn the DM can declare the result of a d20 die roll the creature makes instead of rolling the die. This can be an attack roll or an ability check, including a skill check or a saving throw. If the creature has advantage or disadvantage on the roll, the Legendary Destiny determines the result of one of the rolls, and the other die must still be rolled. It cannot replace rolls made by other creatures. Once used, a specific Legendary Destiny cannot be used again until the start of the creature's next turn.
Most legendary foes have just a single Legendary Destiny, but some very powerful ones could have multiple uses. For example, Legendary Destiny 10 gives the creature a die roll of 10 to use once at some point in a round. Legendary Destiny 10/10 gives the creature two die rolls to use (both 10s in this case). A very powerful creature might have Legendary Destiny 12/10/9 for three rolls to use over the course of a round.
Thoughts and Notes
Yes, this is a souped-up mash-up of the Diviner Wizard's Portent ability and Reliable Talent thrown in. And the numbers given will require more thought, adjusted for the creature's bonuses and the players' capabilities, but the idea is that the number is a dial that can be adjusted for various threat levels.
At first blush, I like that I can use it for more than just saving throws. Currently, Legendary Resistance (LR) solves the frustrating problem of the BBEG failing a saving throw on round 1 and being taken out before they can do anything. But they can still whiff their attack rolls after that. The BBEG still does nothing, they just do it for three rounds instead of one

.
I also like that it can be a calculation for the DM to make, a gambit. Do I save it to use as a saving throw (a la traditional LR), or do I use it to land an attack roll? Or do I make that Perception check to spot that rogue that has been sniping me?
I also like that in some cases, players can try to beef up a saving throw DC or apply penalties to beat the opponent's LD and force a roll, something they cannot do to beat LR.