Italian could be an option. I've thought about it. It could have happened many ways. Western part of Hungary was under Roman rule for almost 300 years, Pannonia province. My family is almost exclusively has its origins from Western Hungary. During the Middle Ages many Italians came to Hungary, mercenaries, artists, priests, masons etc. King Matthias Corvinus had an Italian wife and a great renaissance court. ("With his patronage Hungary became the first European country which adopted the Renaissance from Italy." en.wiki -
http://www.hup.harvard.edu/catalog.p...=9780674063464) During my search in the National Archaives I've found Bakodi named guys who were part of the royal court, but I couldn't link those Bakodis with my family, because I could only go back till 1593.
But for other options I can come up with planty of theories, like Greece. Hungary had many connections with the Byzantine empire, and was a refugee after Ottoman conquest. Also during Ottoman times many Turkish, Anatolian and people from the Balkans came to Hungary. I also have a distant Armenian match on Y-DNA (10/12, 23/25, 33/37, 57,67), and many Armenians came to Hungary as well. Although Armenians are really a mix, they also had Roman and Byzantine connections. As for other Middle Easterners during the Middle Ages the financial elite of Hungary was primarly Muslim and Jewish. There were many so called "Ismaelite" merchants in Hungary. They even had mosques and separated quarters of cities.