Why do Greek people call their grandmothers “Yaya”?

Because that’s the Modern Greek word for grandmother. 🙂

The Triantafyllidis dictionary gives a shrug for the etymology: Λεξικό της κοινής νεοελληνικής

λ. νηπιακή: γιάγια και μετακ. τόνου για προσαρμ. στα άλλα ανισοσύλλαβα ουσ.

Baby talk: yáya and accent shift to adapt to other imparisyllabic nouns

Babiniotis’ dictionary gives the same shrug.

The motivation is wrong: yaya didn’t have to be imparisyllabic to begin with (and váya, the Mediaeval word for nurse, wasn’t). The obvious analogy is instead with other child-talk terms: mamá “mum”, babás “dad”, papús “granddad”, dadá “nanny”.

Baby talk has given us mama, papa/baba (hence babás “dad” helped by Turkish), and dada (hence dadá “nanny” again from Turkish). I’m not aware of yaya as an established baby-talk vocable, but I don’t see what else it could be.

Another, now obsolete word for grandmother btw is nené: νενέ – Wiktionary. That’s also from Turkish, and it also fits the baby-talk pattern.