Can Daughter of the Void (Kai'Sa) pay for extra costs of abilities created by spells like Falling Star or Icathian Rain?

Ruling: No, Kai'Sa cannot pay for extra costs of abilities created by spells. These abilities are part of resolving/finalizing a trigger, not part of playing the spell itself.

Nuances:

• Kai'Sa can pay for Deflect costs on spells because Deflect changes the cost you pay for the spell itself
• There is no rule that gives "responsibility for costs" to the game object that created an ability
• When you pay extra costs for triggered abilities, you are just finalizing the ability, not paying a cost attributed to the original spell