Firstly, we don't
know any Indo-European root words - we can guess at the possible shape of some of them by comparing similar words across the various languages, but we don't actually know what any of them sounded like.
According to my research, Iran is a modern version of Aryan, referring to the ancestry the Iranians lay claim to. Both of these come from the Sanskrit term arya, which does mean noble. The name
Aryana comes from this, but while its meaning may be derived from arya, its given meaning is "from the land of the Aryans", ie "from Iran".
So yes, I'd say
Aryana (which could also be spelled
Ariana) and the
Ariadne Ariana are just coincidental forms.
My reference:
http://www.wordiq.com/definition/Arya
Most names which mean "noble" in the BtN database are from the Germanic root word
adal. Also,
Adara means "noble" in Hebrew and
Alina means "noble" in Arabic.
Hope that helps!