Yes, I agree that Greek names travelled all over Europe, but the names that travelled were by and large those well-known from mythology or history. I studied Greek Mythology for one year and Greek Archaeology/Art and Architecture for onr year, and I never heard of an Ada, so it seems she was an obscure character whose name might not have the same travelling power as, say, Cassandra or Alexander.
I didn't mean quite what you thought with the "set of syllables" thing - all names are sets of syllables, whether those syllables have independent meanings or not. What I meant was that the two syllables 'ay' and 'da' are found in most languages, so it's not surprising that the combination of the two as a feminine name is found in more than one language.
I didn't say anything about Ayda - AY-da is the phonetic transcription of the pronunciation of Ada.