I know some Daughertys who pronounce it DOH-er-tee.
The predominant response in all of these seems to be: the correct way, and the American way.
It’s very amusing for me to hear what you’ve done to all the French names in the U.S.
I used to work with two brothers whose last name was DiRienzo. And they pronounced it differently. One pronounced it as three syllables - di-renz’-oh - and one pronounced it as four syllables - dee-ree’-enz-oh.
Well you know what they say, one man’s fish is another man’s poisson.
Among my many faults is that I’m a Cleveland Indians baseball fan. It drive me nuts that Indians executive Mark Shapiro pronounces his last name as shap-eye-ro, rather than shap-eer-ro.
It bugs me more that he’s a lousy baseball executive but that’s another rant.
My old college girlfriend married a guy named Ruiz who pronounces it reese. I asked her how she could marry a guy who can’t even pronounce his own name properly but she just smiled and said that wasn’t important. 
Hmm. Considering that most people here - NH has the highest French pop. in the country - whose last names are French pronounce them in a totally different way than they’re spelled, I wonder how close they have kept them to the orginial pronunciations. They’re roughly things like: BOUCHARD = Boo-shard, MICHAUD =Me-shue,
PROULX = Prue, DUBE = Do-be…
Apparently one of the names in my family tree can be pronounced two ways. It’s Faunce, which rhymes with an explorer’s name (you know, Ponce). The other way of saying it is the same as the word fonts, without the “C” sound at the end. A few of the more distant relatives favor the latter. I believe at one point it was actually spelled more like Fonts, though.
I met someone recently whose last name is McLeay, and to my surprise he says "Muh-clay, rather than Mick-Lay - the few people I’ve met before with the last name have said the latter. Does anyone know if it’s an Irish or Scottish surname, though? Maybe it’s both, which would account for the difference.
I know a guy in high school named Martinez.
Every other time in my life I’ve heard that name it was pronounced “mar-TEEN-ez,” but his immediate family was “mar-tin-EHZ.”
I’ve known people who have pronounced it that way too, but not as many as the other way.
And to add to the confusion, there’s Leslie Favre, a former Cleveland-area high school basketball player who pronounced (and probably still pronounces) her surname like the word “favor”.
Meanwhile, ex-NBA baller Mike Riordan favored REAR-don.
It’s the same in a lot of Vermont (#2 for percentage of French-Americans.) Hell, my family name is one of the ‘bastardized’ pronunciations. We know the proper French way to pronounce it, we just don’t care. 
That’s how you pronounce Proulx in French, more or less - the lx is silent. Same goes for Groulx.
A branch of my family has the last name Koppel. Instead of pronouncing it like the reporter Ted Koppel does, they pronounce it “koh- PELL” with the stress on the second syllable.
I know someone with the name Michele who pronounces it MISH-lee
And a man who pronounces his last name -Stone- like it has 2 syllables slurred together: Stow,own.
Everyone who knows who they are knows that Nanette Fabray is the aunt of Shelley Fabares
Floyd Robertson: And now to our reporter at the scene, Ms. Tawny Beaver
Tawny: That’s Bee-Vay!
I knew someone in high school whose last name was spelled the same as that of a local politician, but pronounced differently. One day he told us that up until his grandfather’s generation, his family had used the politician’s pronunciation. Then his grandfather had a fight with his family, stormed off, and changed his name.
But only the pronunciation, not the spelling.
(I don’t know why I think telling you this guy’s name would lead people to identifying my real life identity, but I’m not telling you this guy’s name anyway. One pronunciation rhymed with “weigh” and one with “thigh”, roughly)
You don;t pronounce the ‘l’ as well? Bother. I’ve been doing it wrong.
There’s a person we at work know whose last name is “Menzies”. I read his name off a sheet as “Men-ish”, which led to a lot of blank stares. Here in the US it’s pronounced as it sounds (Men-zees), but in Scotland, it’s pronounced “Men-ish”, as in Menzies Campbell, the LibDem politician.
Yet another person we at work know had a mother whose maiden name was “Rougeaux.” Happily, I pronounced it correctly as “Roo-gie.” It turns out that his mother and my grandmother, another Rougeaux, were 2nd cousins.
Well, if those goofy French wouldn’t insist on pronouncing things totally different from the way they’re spelled…
I worked with two Desmaris’. One pronounced it Da-mare-is. The other pronounced it Deh-meh-ray.
My boyfriend used to work with a woman named Meghan. She insisted it was pronounced Meegan. She also insisted that the military planes constantly flying overhead were UFOs (this was about 10 miles from an AF reserve base).