Maybe tie it into dexterity. Like if you have high dexterity you move faster so you get your turn faster than others.
That ... is exactly what Dexterity already does?
I don't think that many PHB Classes or Subclasses have anything that gives a bonus to Initiative rolls, and I don't believe there are many spells that do that either. But maybe we will find some magic items that give us a flat bonus or maybe advantage on Initiative rolls.
