I agree that the "princess" meaning is probably wrong.
I've checked a online Greek dictionary, and the words in Greek for "princess" appear to be prigkípissa and vasilopoúla. Neither looks anything like
Tia or
Tiana. I also checked an Irish Gaelic dictionary, and none of the words for "princess" resemble
Tia or
Tiana.
There was however an Ancient Egyptian princess called
Tia:
http://en.wikipedia.org/wiki/Princess_TiaHowever I can't find any source for the Ancient Egyptian meaning of the name.