David M. Bingaman (1842-1896), served in the Civil War in Companies C, D, and E of the 20th Indiana Infantry. Family lore has it that he was wounded in action at the Battles of Malvern Hill (1 Jul 1862) and Gettysburg (2 Jul 1863). He survived these wounds, but older brother, John M. Bingaman, whom David followed into the Army, perished in combat at Malvern Hill, Virginia. David went on to marry Amanda A. McKibben in 1871. They lived in Illinois, the Oklahoma Territory and Kansas. The couple had no children.
As a deceased Union Civil War veteran, his grave in Pomona, Kansas was marked with a headstone supplied at government expense in 1902, under legislation passed in 1879 (20 Stat. 281). Besides the allowance for grave markers for Union veterans in private, village and city cemeteries, the law stipulated
The Secretary of War shall cause to be preserved in the records of his Department the names and places of burial of all soldiers for whom such headstones shall have been erected by authority of this or any former acts. 
Today, headstone records for interments in private cemeteries for the period between 1879 and roughly 1903 are part of Record Group (RG) 92 Office of the Quartermaster General. Per the catalog entry there are 166,000 cards that have been microfilmed on 22 rolls. The microfilm may be accessed at the National Archives in Washington, D.C. or at regional branches of the National Archives. Nine 3” x 4” inch cards were microfilmed per frame. This microfilm collection has also been digitized, and is available at Ancestry.com as Headstones Provided for Deceased Union Civil War Veterans, 1879-1903.
Information from the card is as follows:
Name: Bingaman, David M.
Rank: 2nd Lt.
Service: Co. D, 20th Regt., Ind[iana] Inf[antry]
Cemetery Location: Pomona, Franklin Co., Kans.
Date of Death: Nov 30 – 1896
Headstone Supplied by: Lee Marble Works
Contract Date: March 29, 1902 
I have not yet been able to ascertain whether applications for headstones made between 1879-1903 might exist, although I have seen earlier examples online at NARA, and catalog entries for the period following. This will be added to my to-do list when I attend the National Institute on Genealogical Research (NIGR) in Washington, DC in July 2013.
Read more about this topic:
Kluskens, Claire Prechtel. “Headstone Records for US Military Veterans, Part II: Records for Headstones Requested from 1879 to 1925.” NGS Magazine 39:2 (April-June 2013), 32-35. A copy of this article may be downloaded by NGS members at http://www.ngsgenealogy.org.
Mollan, Mark C. “Honoring Our War Dead: The Evolution of the Government Policy on Headstones for Fallen Soldiers and Sailors.” Prologue 35:1 (Spring 2003), 56-65. Online here.
 “An act authorizing the Secretary of War to erect headstones over the graves of Union soldiers who have been interred in private, village, or city cemeteries,” 20 Stat. 281 (3 Feb 1879).
 “Headstones Provided for Deceased Union Civil War Veterans, 1879-1903”, card for David M. Bingaman (1902); digital image, Ancestry.com (http://www.ancestry.com : accessed 28 May 2013), citing NARA microfilm publication M1845, roll 2.