When people ask me where I'm from, I usually say that I was born in Florida but grew up in Georgia, even if I know my new acquaintance wants to know where my name came from.
Most usually, I have to clarify by saying, "Both of my parents are from Iran." And when I say "Iran," I don't pronounce it "I-ran," but "Ee-ron," as Iranians (read: Ee-ronians) would.
(Saturday Night Live Digital Short, I Ran So Far)
Although it's incorrect in Farsi (the national language of Iran), I know it shouldn't bother me to hear people say I-ran. After all, nobody says La France when talking about France in English, or España when talking about Spain. I once was very lightly chastised by another Iranian-American for not introducing myself as SAA-naaz (the "a" here sounds like the "o" in "opera"), but opting for the English friendly pronounciation Sun-OZ. But just as names of countries necessarily sound different in different languages, so too do names of people.
Even the distinctions of "Persian" and "Farsi" are mixed up in histories of other languages. The title "Persian" comes from the Greek Persis (then the Latin Persianus), from the ancient Iranian name for the country, Parsa. Sometime after the Arab invasions of the seventh century, this was changed to "Farsi" because there is no "p" sound in the Arabic language. For this reason, I'm fairly certain that "Persian" or "Parsi" is the technically correct term for the language (although I and most other Iranians I know designate it as Farsi).
If people ask me where my name is from directly, I usually respond that it's Persian. I don't know why I would necessarily expect anyone to know what that means. The first time I heard that someone was Guamanian I had to ask for her to repeat it (well, it's less obvious when spoken quickly--written it's more direct), so it would stand to reason that some people wouldn't necessarily know to connect Persian with Iran. Usually, I'm impressed when I tell people that my parents are from Iran and they respond with, "Oh, so you're Persian."
A lot of people ask me if Farsi is "like" Arabic. Although several words in Farsi are borrowed from Arabic, and the language is written with the Arabic alphabet, the languages are actually very different; unlike arabic, Farsi is an Indo-European language (basic words like mother--maadar in Farsi--and father, or pedar in Farsi, are indicators). Similarly, Persians and Arabs are ethnically different, a point many Iranians (who are in general a people very proud of their ancient culture and bitter about the still relatively recent Iran-Iraq war) would likely be quick to point out. Iranians are ethnically Aryan, and didn't share the same nomadic herding lifetsyle attributed to early Arabs.
(Comedian Maz Jobrani, from the "Axis of Evil" Comedy Tour)
The linguistic overlap and entymological changes aren't exceptional by any means. The language of American English also has borrowed many words and phrases from other languages, and is misprounounced in other countries (as EEN-glee-see in Farsi, for example).