The term “old college try” was popularized in baseball, as a fielder gamely makes a try to go after the ball. “Old college try” usually describes a futile effort, with the player expending energy in an impossible task.
 
“When Wagner’s ball soared out to centre on its long flight the Boston players danced with glee and joyous abandon. It appeared that there was not a possible chance for the ball to be caught, and Stahl started to pound around the bases with his run. But Snodgrass gave it the old college try and got away with it” was printed in the Buffalo (NY) Commercial on October 16, 1912.

New York Giants baseball manager John McGraw, in a July 1913 newspaper article about managers not thinking too highly of college educated ballplayers, was quoted:
 
“Ah, well,” said McGraw, wearily, as the youth came back to his bag, “he gave it the old college try.”
 
A 1917 “The Old College Try” sermon by the popular preacher Billy Sunday (1862-1935) also credited John McGraw.
 
 
Wiktionary: old college try
Noun
old college try (plural old college tries)
1.(idiomatic) A vigorous, committed attempt or effort.
1929, Munsey’s Magazine, vol. 96, no. 3, p. 355:
Those who have seen Ruth make the “old college try” understand that some professionals play with a spiritual fervor which is supposed to be the amateur’s prerogative.
Usage notes
Often used in the expression give it the old college try.
 
Free Merriam-Webster Dictionary
college try noun
Definition of COLLEGE TRY
: a zealous all-out effort
Origin of COLLEGE TRY
from the phrase “give it the old college try”
First Known Use: 1927
   
Wikipedia: John McGraw
John Joseph McGraw (April 7, 1873 — February 25, 1934), nicknamed “Little Napoleon” and “Muggsy,” was a Major League Baseball player and manager. Much-lauded as a player, McGraw was one of the standard-bearers of dead-ball era major league baseball. Known for having fists as quick as his temper, McGraw used every advantage he could get as both a player and manager. He took full advantage of baseball’s initial structure that only provided for one umpire, becoming notorious for tripping, blocking and impeding a baserunner in any way he could while the umpire was distracted by the flight of the ball. His profligacy in employing such tactics may have led to additional umpires being assigned to monitor the basepaths.
 
However, even with his success and notoriety as a player, he is most well known for his record as a manager. His total of 2,763 victories in that capacity ranks second overall behind only Connie Mack; he still holds the National League record with 2,669 wins in the senior circuit. McGraw is widely held to be “the best player to become a great manager” in the history of baseball.
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Teams
As Player
Baltimore Orioles (AA/NL) (1891–1899)
St. Louis Cardinals (1900)
Baltimore Orioles (AL) (1901–1902)
New York Giants (1902–1906)
 
As Manager
Baltimore Orioles (NL) (1899)
Baltimore Orioles (AL) (1901–1902)
New York Giants (1902–1932)
 
Newspapers.com
16 October 1912, Buffalo (NY) Commercial, “Snodgrass Has Redeemed Himself,” pg. 5, col. 4:
When Wagner’s ball soared out to centre on its long flight the Boston players danced with glee and joyous abandon. It appeared that there was not a possible chance for the ball to be caught, and Stahl started to pound around the bases with his run. But Snodgrass gave it the old college try and got away with it.
     
Newspapers.com
5 July 1913, Lancaster (PA) Daily Intelligencer, “News from the World of Sports,” pg. 6, col. 2:
McGraw one time had a college first baseman on trial. A foul was hit a mile over the grandstand, but the collegian tore after it, cap gone and hair flying. “Ah, well,” said McGraw dryly, as the youth came back to his bag, “he gave it the old college try.”
 
24 August 1913, Kalamazoo (MI) Gazette, “College Education No Aid on Diamond, Declare Pilots,” pg. 11, col. 2:
McGraw once had a college first baseman on trial. A foul was hit a mile over the grandstand, but the collegian tore after it, cap gone and hair flying.
 
“Ah, well,” said McGraw, wearily, as the youth came back to his bag, “he gave it the old college try.”
 
31 May 1915, Oregonian (Portland, OR), “Seals Nick Higg For 2 Home Runs” by Roscoe Fawcett, pg. 10:
This seemed to take all the “old college try” out of Hig’s right propeller, and he never entirely recovered his composure until squatted under the shower baths.
   
Newspapers.com
17 June 1917, Sioux City (IA) Sunday Journal, pg. 18, col. 3:
“The Old College Try”
By BILLY SUNDAY
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But McGraw, Wiser than Any of them, said:
 
“That the Eye, Young Fellow—The Old College Try!”
 
And ever since then, when an Ambitious Athlete goes after a Hard One, the Giants call it “The Old College Try.”
 
“They Miss Most of Them, don’t they, Mac?” somebody asked him.
 
“Yes,” answered the Cagey Chieftain—“but it would Jolt You to see how Many they Grab!”
 
Chronicling America
20 August 1922, New York (NY) Tribune,“Yankees Trounce White Sox, 12-5, and Gain Clear Title to First Place” by John Kieran,  pg. 14, col. 1:
A looping liner by Bib Falk that Ruth gave the old college try, resulted in a homer for the White Socker in the second, and the score was tied for a brief period.