Frank Lloyd Wright - Wisconsin
Heck, I’m from Arkansas, and I had to look up the guy. Of course, I’ve been on record as pointing out that I don’t really follow sports. I might watch a game with friends, but all I do is pick a team and hope it wins.
With Ohio, you’ve just got so many. Edison is a good choice, but with Armstrong, Glenn, the Wright Brothers.
Then there’s EIGHT US presidents. Now, while several are the ‘nobodies’ of presidents (William Henry Harrison, Hayes, etc…) Grant is pretty universally recognized, as is Taft.
Then there’s others like Annie Oakley, Jack Nicklaus, Steven Spielberg, Cy Young…
However, if we’re going for FAMOUS…it’s got to be LeBron James. I guarantee, worldwide right now, James will have the most photo/name recognition of any Ohioan.
Cambridgeshire:
Douglas Adams
John Maynard Keynes
Oliver Cromwell
(Although, probably not as famous internationally.)
Jim Thorpe was born in Oklahoma. He’s just buried in Pennsylvania.
Malcolm X was born in Nebraska.
New Hampshire:
Alan Shepard
Dan Brown
Sarah Silverman
Salmon P. Chase
Franklin Pierce
Daniel Webster
I’d guess today, Dan Brown or Sarah Silverman are the most famous.
Just a little idea for the OP to consider:
Once the thread has had at least one entry for each state by someone identified as being in that state, and once there appears to be some reasonable dispute over who is the “most famous” for a state, simply list the ones in contention in a poll and let Dopers decide who is most famous.
After all, it is an SDMB thread under study. International fame is a good criterion, but we can’t easily poll international opinion here.
States like Massachusetts, Ohio, Texas, and even Tennessee (just to name some) already have some reasonable contentions going. Separate polls by disputed state would be fun for all of us to say who is famous and who isn’t.
As I say, just an idea. I’m thinking of starting such a poll for Tennessee based on my pick of maybe 20 from the list at Category:People from Tennessee - Wikipedia but I would gladly leave that exercise to the OP of this thread for consistency’s sake.
You have Helen Keller for Alabama. I would like to propose Jesse Owens.
While not all the presidents are well known to everyone, if you want them on the list you have to add from Ohio:
James A. Garfield
Ulysses S. Grant
Warren G. Harding
Benjamin Harrison
Rutherford B. Hayes
William McKinley
William Howard Taft
Other famous Tennesseans:
Sgt. Alvin York
Aretha Franklin
Bettie Page
Cybill Shepherd
Dale Dickey (Patty the Daytime Hooker on My Name is Earl) 
Debbie Allen
Dixie Carter
Dolly Parton
Greg Allman
Isaac Hayes
Jack Daniels
Justin TImberlake
Kathy Bates
Kenny Chesney
Miley Cyrus
Morgan Freeman
Reese Witherspoon
Rita Coolidge
Tennessee Ernie Ford
Tina Turner
StG
Too late, but I came to mention Jack Daniels and Bettie Page for Tennessee. I mean, what else does one need in life!
Missouri. Mark Twain comes to mind.
I’ve always been interested in how a state “claims” a celebrity or famous person. Walt Disney is touted as being a famous Missourian, however he was born in Illinois. Also, Daniel Boone was a Missourian, but he was born in Kentucky (I think). Brad Pitt is another example. I’ve read many interviews and he says he is from Missouri, yet he was born in Oklahoma. Scott Joplin was born in Texas, but is another famous Missourian.
On the other hand, Eminem was born in Missouri…
South Dakota
Today, probably Tom Daschle or Tom Brokaw, although Hubert Humphrey and George McGovern were also born here. Other contenders include Harvey Dunn, Cheryl Ladd, Terry Redlin, and Sitting Bull.
Virginia has Patsy Cline, Thomas Jefferson, Robert E. Lee, James Madison, Warren Beatty, Wayne Newton, Patton Oswalt. How famous each of those people are to you depends on your interests.
Disney was born in Chicago.
Illinois:
John Deere
Hilary Clinton
Miles Davis
Carl Sandburg
Harrison Ford
Jack Benny
Bob Newhart
Jane Addams
Henry Wadsworth Longfellow
Dorothea Dix
Margaret Chase Smith
Fair enough. I withdraw Walt Disney and replace him with Walter Cronkite.
Aslo, Chuck Berry.
Wyoming:
Buffalo Bill Cody
Jackson Pollock
Dick Cheney
I thought about going with Jack Dempsey (aka “The Manassa Mauler”) as well, but wondered how famous he is nowadays. That’s debatable, I suppose.
Neither Glenn Miller (born on a farm in Iowa) nor Margaret Brown (Hannibal, Missouri – the boyhood home of Sam Clemens, as it turns out, but not his birthplace) were born in Colorado.
Cody was born in Iowa (actually the territory that would become Iowa) and Cheney was born in Nebraska.