Why does Kai'Sa have a void name?
Kai'Sa is from Runeterra. She wasn't originally from the void, so unless Kassadin dropped her in as a baby before he named her, it makes zero sense for her name to be of Void origin.
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All void names actually have an ancient Shuriman origin; Icathia was near Shurima, and the Xer'Sai frequently plague there, so it uses ancient Shuriman naming conventions.
So either: A. Kassadin has a thing for ancient Shuriman and gave his daughter an ancient Shuriman name (unlikely) B. She chose the name herself in order to fit in with the rest of the Void creatures and survive (somewhat possible) C. The void creature symbiote suggested it/named her after she forgets her own name (depends on the intelligence of the void creature; void intelligence levels go from voidlings to Vel'Koz, so it's hard to predict) D. The other Runeterrans she meets after emerging from the Void gave her a void name (most likely IMO)