I heard an interesting story about Greek island of Lemnos where people still considered themselves as Roman citizens in 1912.
When the island was occupied by the Greek navy, Greek soldiers were sent to the villages and stationed themselves in the public squares. Some of the children ran to see what these Greek soldiers, these Hellenes looked like. ‘‘What are you looking at?’’ one of them asked. ‘‘At Hellenes,’’ we replied. ‘‘Are you not Hellenes yourselves?’’ he retorted. ‘‘No, we are Romans."
It seems very unlikely that after fall of Byzantine Empire people the island did not have any influence from outside world for more than 400 years. Is there any proof that this story is true ?
I don't know if the incident you are referring to is true or not, but I can imagine that it probably lost something in translation. Greeks do call themselves Ρωμιοί (Romii) and not Ρωμαίοι(Romei). This is indeed a linguistic transfiguration of Ρωμαίοι(Romans) through the centuries. The term is still used to some extend today even though it has fallen out of fashion that past few decades. Under Ottoman occupation the Greek orthodox population was called Rûm or Roum or Rhum which was the Arabic term for Romans. The official administrative name of the Greek mainland(and some of the islands) under the Ottomans was Roumeli.
The term Ρωμιός(Romios) was still in daily use until a few decades ago by Greeks when referring to themselves as a nickname to the Hellenic nation in general. References to this can be found in songs, movies and poems throughout the 50s, 60s and 70s