Armenian surnames typically came from the father's given name, by adding "ian" (son of) to the end of the name. Occasionally, I've seen surnames without the ian added, but that's rare.
A search of Ellis Island records for Avadis brings up very few records - two from Greece and two from Romania. There's also an Armenian from Smyrna listed, but I'm pretty sure they switched his first and last names by mistake. A Google search for Avadis brings up some Armenians and also some data analysis software. No Greeks found in the first few pages of search results.
I'd say there's a pretty good possibility that the Romanians and Greeks are the descendants of transplanted Armenians.