Good to know that some traditions haven’t died - even though I still live in the DC exurbs, and work (just barely) inside the Beltway, I haven’t picked up a CityPaper in many a year.
Hawai’i:
Kamehameha I
Queen Lili’uokalani
Duke Kahanamoku
Dan Inouye
Royalty, Surfin’ and a Hero/Statesman - Hana Hou!
I was born in New York in 1961 and have lived in Texas since 1986. Both states have WAAAAAY more than 4 worthy candidates. But here goes:
New York:
Alexander Hamilton, Washington Irving, Babe Ruth, Franklin Roosevelt
(Too many near misses to name… but Verrazano, Peter Minuit and James Fenimore Cooper are among them.)
Texas:
Sam Houston, Buddy Holly, Lyndon Johnson, Tom Landry
P.S. I KNOW Hamilton and Ruth weren’t originally from New York, and Houston wasn’t orginally from Texas.
And wouldja believe… Naismith is the ONLY man who ever had a losing record as head coach of the Jayhawks basketball team!
Also living in Hawaii, can’t argue with those. I may put Kalakaua instead of Liliuokalani. I was thinking of Don Ho or Elvis, but now I like Duke Kahanamoku better!
I could put Sanford Dole, who helped lead the revolution into democracy, but I’d probably get lynched.
Virginia is too easy for politicians/statesmen, so: Dave Grohl, Roy Clark, Patsy Cline, and Edgar Allen Poe.
I agree with KCB615, and would also take Ralph Waldo Emerson over Nimoy.
And damn, does it hurt to say that.
Illinois:
Abraham Lincoln
Jane Addams
Adlai Stevenson
Barack Obama
Nebraska:
Father Flannagan
Willa Cather
Standing Bear
Johnny Carson
William Jennings Bryan and Buffalo Bill get honorable mentions, but get left off by being on the wrong side of history a bit too often.
I’m leaving Malcolm X and Gerald Ford off this list per the OP’s instructions - both were born in Omaha, but left the state before their first birthday. Likewise Henry Fonda and Marlon Brando - both great artists with Nebraska roots, but neither one really looked back when they left the state in their youth. On the other hand, Johnny Carson grew up here, started his broadcasting career here, and contributed mightily to the state through his philanthropy after he came to national prominence. He’s in.
A fine list, but may I suggest Frank Lloyd Wright instead of Stevenson?
That’s a darned fine list, but might I suggest Charlie Parker instead of Benton?
For Ohio I’m going with Thomas Edison, Bob Hope, Orville Wright, and James A. Garfield.
No Warren Buffett? 
Ethan Allen, Ira Allen, Ben & Jerry.
Vermont, I assume.
In fond memory of all of the nights we harmonized 'til dawn.
Time for the obligatory quote:
[QUOTE=Dave Barry]
The members took turns lighting sparklers and signing their John Hancocks to the Declaration, with one prankster even going so far as to actually write “John Hancock.”
[/QUOTE]
Neil Armstrong
Clarence Darrow
Thomas Edison
John Glenn
Steven Spielberg
And a bunch of Presidents.
If you’ll allow a rookie to nitpick…
Egan’s a fair choice, Laurence is okay, but the other two are questionable.
Ernest Gruening, Charles Bunnell, James Wickersham, Ted Stevens, Bill Sheffield, Elmer Rasmuson, Austin Lathrop, Carl Eielson, Mike ‘Hell-Roaring’ Healy, Frederick Schwatka, General Wild Richardson and, arguably, the infamous Stuart Udall.
My Rushmore would probably be Gruening, Wickersham, Egan and Stevens, with Rasmuson, Eielson, Healy and Richardson sitting in Sitting Bull Statue territory. Udall would once again be burned in effigy.
I think we have a much better claim to Edison. He moved from Ohio as a small boy. Until our Mt Rushmore is made you can come visit the Edison memorial tower. It has a giant light bulb on top. It’s in Edison, New Jersey.
You done stole the code! The list I was going to post. I might want to include some of these but I’m not sure whom I would replace in your group:
Chet Atkins
Shelby Foote
Christina Hendricks – She and Dolly could get all four slots!
Quentin Tarantino
Alvin (Sergeant) York
I looked over the list at List of people from Tennessee - Wikipedia to see who all might qualify, and there are others for sure. I think this group is close to the Rushmore standard, though.