January Jones
May West (spelling it wrong is okay) – Zetterling
June Haver – Allyson – Lockhart
Plenty of blanks to fill in.
If you give a real name it’s okay to add a junk one or two, such as May DeVonner.
I’d say the April, May and Junes are too legion to bother naming examples of. August is a relatively rare name but not unheard of. Of the rest, January Jones is the only month-namer I can think of offhand.
Augustus Caesar, which kind of works both ways since August is named after him.
Well, there’s always Julius Caesar … 
… and his adopted kid, Augustus. Though that’s a title, not a name. 
Edit: ninja’d! In part.
After posting the OP I remembered August Derleth, one of Lovecraft’s contemporaries.
That leaves us with:
February
March
?July Is Julius close enough?
September
October
November
December
Time to google.
So far the closest I’ve gotten is a few character names, and I know they don’t count.* I met someone, once, named September Dawn. I don’t remember her last name. Not sure if that counts, either.
*[One of the Observers in Fringe is named September.
The protagonist in No Rest for the Wicked is named November.
There’s a marvel mutant named December (Winter Frost - X-Nation 2099).]
Nice!
March of Dimes?
We may have to resort to one of the non-July versions, but Julius, Julie and Julian could fill a few pages off the tops of our heads, I should think.
Of course, there’s that old “July Fly and Apple Pan Dowdy” reason for somebody like Marty McFly (Back to the Future, was it?) to get creative with. 
And a simple search reveals it’s Shoo-Fly Pie!
I’m guess Miss January, Miss February, Miss March, etc, aren’t what you were looking for?
Sorry, I got no one that hasn’t already been listed.
I grew up across the street from a family with a daughter named September. Because her last name was also a word, it’s impossible to Google her, although White Pages shows three people with her first and last name, and one is still in the area so could be her.
Too late for edit, but Dave Eggers of A Heartbreaking Work of Staggering Genius fame named his daughter October.
October is the stage name of a British singer. Her real name is apparently October Hamlyn-Wright.
March Fong Eu is a politician from California.
And the four beer barons: August Anheuser Busch, Sr., August “Gussie” Anheuser Busch, Jr., August Anheuser Busch III, and August Adolphus Busch IV.
Fletcher Christian of Mutiny on the Bounty fame named one of his sons Thursday October Christian.
Searching census records turns up a few people named February, such as the fabulously named February Moony of Van Buren co., TN, from the 1870 census. There’s some Novembers as well, including a November Dick, aged 50, in the 1880 Louisiana census.
There is a big joke here.
“December Green” is an awesome name.