For Initiative checks, I like using Intelligence instead of Dexterity (not in addition to, not as an alternative to, but instead of Dexterity.) Thus, initiative becomes less about quick reflexes and more about quick thinking. It also steals a bit of thunder from the most over-utilized ability score in 5E, and gives it to the least-utilized.
I like the idea mechanicly, or atleast the idea that Intelligence needs more attention, but thematically I think it suits Wisdom better. Something that I've had in the back of my mind for a long time is a system where your even breakpoints still give the +1 hit/ dmg/ saving throw bonusses like they always have, but the odd breakpoints give other side-benefits. That way a 11 or 13 in a stat wouldn't feel so useless. Initiative could be tied to this. For example (and i'm just spitballing here):
9 dex: disadvantage on a stealth checks can never be negated by a source of advantage*
11 dex: You can use dex instead of str to determine jump distance
13 dex: +1 initiative
15 dex:+5 feet movement speed
17 dex: Your ranged attacks ignore disadvantage from the prone condition and melee attacks vs incapacitated enemies always crit
19 dex: +1 initiative and (maybe) something extra on top that is useful and thematic for every dex user, with a once per short rest cooldown. Perhaps something like the weapon abilities in Baldurs Gate 3. Or go wild with something like "You get 2 reactions per round". It would be a way to get people to aim harder for ASI's over feats or could serve as a sort of capstone-ability in campaigns that don't reach level 20.
9 Wisdom: You have a 5% higher chance to spawn enemies when you are on watch
11 wisdom: +1 initiative
13 wisdom: **
15 wisdom: +1 initiative
17 wisdom:**
19 wisdom:**
Int could also have get an initiative-tier somewhere. The main idea is that I think it would make sense that what a bonus to initiative represents could come for a part from wisdom, a part from dexterity and another part from intelligence. If we are going with a +5 bonus (from the classic 20 dex) then perhaps it should be 2 from wis, 2 from dex, 2 from int but those are things that can be filled in at will.
*maybe that's harsh or not at all, i dunno, but it's a place where you can, within reason and creativly, punish stat dumping.
** I can't really keep thinking up side-examples to illustrate a point, but i'm up for expanding on the idea if people are interested.