Here I'm proposing that Amma and Ayya are the first two Dravidian words. I'm not sure what was the initial form of these words used by their ancestors in Africa. The day a child called its mother 'Amma' and father 'Ayya' the people somewhere in North-West of India became Dravidians.(see update 1)
Background of Indian population:
I have already mentioned there were three distinct male migrations to India based on three distinct strains of religious worship. The first were tribals. The second were Dravidians who were related to tribals during their years in Middle-East. The third were Indo-Aryans who were related to Dravidians during their years in Southern Central Asia(North-West India, Pakistan and Southern Afghanistan) and were also related to tribals during their years in Middle-East. All these males took up the girls of coastal migration to India killing all their men.
Ayya and Amma in South India:
Amma(mother) has retained her purity in South India all these years. However, Ayya has changed.
Ayya in Telugu: The rural Telugu people still address their father as Ayya. However, more common usage is for the elder and/or powerful.
Ayya in Tamil: Tamils have stopped addressing their fathers as Ayya. Now, it's only for elder and/or powerful.(see update 2)
Ayya in Kannada: In Kannada Ayya in some places has become Ajja which is an addressing term of ones 'grandfather'. As in Telugu and Tamil Ayya is an elder and/or powerful.
Ayya in Malayalam: Ayya has changed to Achha(n) in Malayalam which is an addressing term for father.
Except Tamil, all other major South Indian languages use Ayya to denote either father or grandfather. And it's obvious how it became a respectful term to address any elder and/or powerful.
North Indian Dravidians and Indo-Aryans don't use Ayya to denote either father or grandfather.
Indo-Aryans and Ayya:
Indo-Aryans once they moved from Southern Central Asia to South-Eastern Europe(Kurgan cultural centre)the term Ayya was modified into Arya. During their years in South-Eastern Europe the meaning also changed from father to noble(may be noble father too, 'Arya Putra' most plausibly meant noble father's son. When Darius boasted he's an Aryan, most plausibly it is 'I'm a noble father').
After spending few thousand years in South-Eastern Europe some of the Indo-Aryans migrated back to India. Though the religion that they had brought became a minor because of the dominance of Dravidian priestly class, they succeded in implanting their language in most part of India. However, the continuity with their Dravidian past was ensured in that word Arya, which is , after all, a derivative of 'Ayya'.
1. As Razib of Gene Expression commented 'Amma' might have had earlier origins.
2. Though common addressing term for father is 'Appa' in Tamil(as in Kannada), Ayya can also be used to address either father or grandfather. So all South Indian languages use 'Ayya' to denote either father or grandfather.