“Bigger/More bang for a/the buck” means that someone wants to spend dollars more effectively. The term began in the U.S. military, when the Pentagon told the December 16, 1953 New York (NY) Times that it was seeking “bigger bang for a buck.” “More bang for a buck” was used by January 1954.
 
The saying “bang for a/the buck” has been used in business as well as in government.
       
 
Wiktionary: bang for the buck
Noun
bang for the buck (uncountable)
1.(idiomatic) Efficiency; cost-effectiveness; value.
 
(Oxford English Dictionary)
colloq. (orig. and chiefly U.S.). bang for the buck and variants: value for money, return on an investment; chiefly in more bang for the buck. Cf. buck n.8
Used originally of military spending, esp. on nuclear weapons (cf. sense 2c).
1953 S. Alsop in N.Y. Herald Tribune 21 Dec. 18/6   They believe that the ‘more bang for a buck’ theory is an excuse for the cutbacks, rather than a real reason, and that the ‘buck’ came first by an easy margin, with the ‘bang’ a poor second.
1976 National Observer (U.S.) 30 Oct. 2/4   The labor-backed Health Security Act provides more bang for the buck—more new services for each dollar of added cost.
 
New York (NY) Times
16 December 1953, New York (NY) Times, “Military Cuts Stir Debate” by Hanson W. Baldwin, pg. 20:
The struggle to get a bigger bang for a buck—or more combat effectiveness for less money—passes a major milestone this week.
 
Google News Archive
21 December 1953, Youngstown (OH) Vindicator, pg. 10, cols. 4-5:
Defense Asks “Bigger Bang for a Buck”
By Stewart Alsop
(...)
Yetthe question remains: Is the “bigger bang for a buck” theory—the theory that nuclear firepower can achieve big money and manpower savings—really sound?
 
Google News Archive
17 January 1954, Tuscaloosa (AL) News, “Main Question Raised By Dulles’ Plan For Military Strategy: Will It Work?” by James Marlow (AP), pg. 6, col. 3:
“Maximum protection,” he said, at a “bearable cost.” This has been referred to in Washington as “more bang for a buck.”
 
OCLC WorldCat record
More bang for the buck.
Author: Bennie Jew
Publisher: Maxwell Air Force Base, Ala., Air War College, Air University, 1970.
Series: Air War College research report, no. 4041. 
Edition/Format:  Book : National government publication : English
   
OCLC WorldCat record
Bang for the buck! : how to pay people for the performance you want
Author: Dale E Melin
Publisher: Grand Rapids, Mich. : Capture Communications Division, Capture Corporation, ©1992.
Edition/Format:  Book : English