Why is Zyra spelled with a Z but Xayah is spelled with an X

Because Xerath was getting lonely

You mean because they are kinda pronounced the same?

The explanation I personally came up with: Place of origin of their names. Zyra is from the Kumungu, while Xayah is from Ionia.

Ionia as a region is heavily inspired by Asian culture. And the pronounciation can differ alot from the western way here. Take (for example) Chinese names like Xi Rugu. While, if transcribed in latin letters, the name gets written with an X, the pronounciation is different. You can listen to an example how to pronounce this name here.

So, I would say that the reason is cultural difference. Riot is putting alot of work in the lore for some time now and are thinking deeply about stuff like the languages used in their different societies. I suspect that they made use of this differentiation when it comes to names, too.