Where does “天草” come from?

天草 (Chinese) comes from Japanese 天草, from Japanese 天, from Old Japanese 天, from Proto-Japonic *amay.

Ancestry of “天草”, step by step

Step	Language	Word	Meaning
1	Japanese	天草	any of the red algae of the genus "Gelidium";...
2	Japanese	天	the sky, the heavens; the area above a kitchen...
3	Old Japanese	天	sky, heavens
4	Proto-Japonic	*amay	—