Welcome, Johnny!
A question you need to determine is: Is your last name really Miconi? Or is that what the immigration officer who processed your immigrant ancestors thought he heard when they arrived in this country? A typical example is seen in GFII, when the boy Vito arrives in America. The immigration officer asks, "What's your name, son?" An Italian-speaking assistant says to Vito, "Uo nome?" Vito doesn't answer, so the assistant looks at his papers and says, "Vito Andolini from Corleone." The immigration officer say, "OK, Vito Corleone. Next..."
That kind of stuff actually happened all the time. My real family name is polysyllabic. But when my grandfather came to Ellis Island around 1900, the immigration officer recorded only the first syllable--and that's been our name ever since.
In your case, suppose your ancestor's name was "Micchione." He/she would pronounce it "Mih-CONE-eh," and the phonetic version would go on the immigration certificate. And that's only one possibility. You need to quiz your older relatives to find out more.