Most Irish names the spelling might look weird if you're thinking English spelling, but the pronunciation isn't all that different. You can spell my first Kevin or Caoimhghen or a dozen other ways, but the pronunciation stays the same - well the vowels can vary according to the accent, but then they do with the English spelling as well. And the same would go with Meehan.

That may not be so true in America as in England - my impression is that Americans are more likely to change the way a foreign name is pronounced to fit in with conventional orthography, as with a name like Wagner. So if you aren't in an Irish community, I suppose the pronunciation of the name might have drifted away from the way it started out.