Hamilton “Ham” Coolidge was born on September 1, 1895 in Brookline, Massachusetts to Professor J. Randolph Coolidge, Jr. and Mary Hamilton (Hill) Coolidge. He attended Groton School (Class of 1915) and was Senior Prefect, captain of the football team and a pitcher on the baseball team at the school. He then attended Harvard University with his close friend Quentin Roosevelt, the youngest son of President Theodore Roosevelt. Coolidge was Vice President of the freshman class, a member of the freshmen baseball team, and a member of the 1916 varsity football team while at Harvard. Coolidge would attend various flight training camps over the summers during college; in 1915 he attended a Reserve Officers' Training Corps camp in Plattsburg, New York and in 1916 he attended the Curtiss Fly School in Buffalo, New York.

When it started to look increasingly likely that the United States would be entering World War I, Quentin Roosevelt and Coolidge, along with their friend Douglas Campbell, all dropped out of Harvard and applied to join the Army. Coolidge officially entered United States service in March of 1917 at Key West, Florida, where he enlisted in the Aviation Section of the U.S. Signal Corps. Once the United States officially entered World War I, Coolidge was sent to ground school at the School of Military Aeronautics at the Massachusetts Institute of Technology, and graduated on June 5, 1917.

The Aviation Section of the U.S. Signal Corps (which eventually became the U.S. Air Force) was full of young men from wealthy families looking for a bit of adventure, so Coolidge and Quentin Roosevelt were not alone in their persistence to join the war effort. The so-called "Millionaire's Unit" that flew for the United States Navy in World War I was a similar outfit, though the men in it were from Yale instead of Harvard.

Coolidge embarked for France on July 23, 1917 and was initially stationed at the American Expeditionary Force Headquarters in Paris, France. He received his commission as a 1st Lieutenant on September 29, 1917 and was sent to the 3rd Aviation Instruction Center in Issoudun, France. He served as a squadron commander and then transitioned into a role as a test pilot, flying 8 to 10 times a day.

Coolidge made it to the front when he was assigned to the 94th Aero Squadron on June 16, 1918. The Squadron was nicknamed "The Hat in the Ring Gang" to commemorate the United States entering World War I and "throwing its hat in the ring." Their insignia consisted of Uncle Sam's red, white and blue top hat going through a red ring. The pilots in the squadron became some of the most decorated flyers of the war, with six pilots earning the Distinguished Service Cross and five men earning Ace status. As a group, the pilots downed 70 enemy aircraft during the war.

Coolidge participated in many missions while stationed in Toul, France with the 94th and was promoted to Captain on October 3, 1918. On October 27, he was killed in action when his SPAD XIII (S.13) took a direct hit from an anti-aircraft shell near Grandpré, France. Fellow pilot Eddie Rickenbacker and a priest rode out to the spot, which was just a few yards inside their lines. They were able to recover the body and give it a proper burial. Rickenbacker made note of the ordeal in his memoirs:

Early next morning I secured a Staff car and proceeded up to the front to find the spot where lay the last remains of my dear friend. We reached Montfaucon and turned northwest around the edge of the Argonne Forest, passing on the way the wreckage of my red-nosed Fokker just outside the town of Exermont. Arrived to within a mile of our front line, sheltered all along the road by hanging curtains of burlap and moss, part of which had been left by the Huns and partly our own concoction of camouflage, we were halted by an officer who told me we could move no further without coming under shell fire from the enemy guns.

Abandoning the car at the roadside, we skirted the edge of woods that adjoined the road and made our way on foot to the flat lands just across the Aire River from the opposite town of Grand Pre. And here in the bend of the Aire, almost in full sight of the enemy, we came upon the body of Captain Coolidge. A lieutenant in infantry who had seen the whole spectacle and had marked down the spot where Ham's body had fallen, accompanied us and it was through his very kind offices that we reached the exact spot without much searching. The Chaplain of his regiment likewise accompanied us. And there, not sixty yards behind our front lines, we watched the men dig a grave. The Chaplain administered the last sad rites. Amid the continuous whines of passing shells we laid the poor mangled body of Captain Hamilton Coolidge in its last resting place. Over the grave was placed a Cross suitably engraved with his name, rank and the date of his tragic death. A wreath of flowers was laid at the foot of the cross. Then with uncovered head I took a photograph of the grave, which later was sent "back home" to the family who mourned for one of the most gallant gentlemen who ever fought in France (From pages 342-343 of Fighting the Flying Circus by Edward V. Rickenbacker, Stokes Company, 1919).