The Oxford English Dictionary (2^nd edition, 1989) defines backroom boy as denoting:

– a person engaged in research, especially of a secret nature,
          hence also, in extended use:
– a person who works or wields influence ‘behind the scenes’.

The primary definition is due to the fact that this dictionary records as the earliest quotation the following from The Listener (British Broadcasting Corporation) of 27^th March 1941, which reported that the Canadian-born British Conservative politician and newspaper proprietor William Maxwell Aitken (1879-1964), 1^st Baron Beaverbrook, had declared, as Minister of Aircraft Production:

Now who is responsible for this work of development on which so much depends? To whom must the praise be given? To the boys in the back rooms. They do not sit in the limelight. But they are the men who do the work. Many of them are Civil Servants.

But boys in the backroom, or backroom boys, appeared in the 1920s as a U.S. political term denoting persons exercising a surreptitious influence.

The earliest instance that I have found is from an article about the result of the Republican Party primaries in Minnesota, published in The Washington Herald (Washington, D.C.) of 26^th March 1920:

Independent Vote Big.

Regardless of the hybrid nature of the Minnesota primaries and the apparent “set-up” for one candidate, the results are giving the managerial talent in all camps considerable food for reflection. Mathematically figured, only 12½ per cent of the voters had a physical chance of getting near the ballot boxes in the time allotted. Despite this fact there was a marvelous amount of “pasting” something, which positively had not been looked for. Stickers disfigured the ballots and the scrawls gave the “regulars” something to think about. In the opinion of neutral observers here it proved conclusively that anyone who can forecast the results next fall will indeed be a prophet.
The thousands of “irregular” votes in the much restricted primary made it patent that the “controlled” vote is a thing of the past and that the great majority of intelligent citizens are going to think for themselves.

Make Bid to Labor.

The warning of the voters in Minnesota, the few who had a chance to register their real preferences, is a portend of what may happen should the “boys in the backroom” meet at the zero hour to pick the candidate. It is expected that in succeeding primaries equally significant “preferences” may come to the surface. For this reason as well as for others violent efforts are being made by practically all the candidates to show a more liberal spirit than has been manifest up-to-date. More and more the principal figures are appealing to labor and judging from the numerous protestations there is not a candidate who is not lying awake nights thinking of the welfare of the laboring man.

The second-earliest instance that I have found is from The Great Bend Daily Tribune (Great Bend, Kansas) of 28^th February 1922 (I have not corrected the numerous mistakes in the text):

TAXATION BUT NO BENEFITS

Increases in taxation are always excusable when the expenditure of money means anything to the community or state. If the city or state, or nation is building, if it is making things more attractive, if it is getting more it is worth whole to spend more. It would be an investment and and taxes increased would be excusable. Right now the talk of taxpayer’s leagues is liable to influence many to a point that they will overlook real returns—real investment. A tapayers league which decries immovement can be nothing but a political movement and no matter who is at the head the political boys in the back-room will be running it. A rax payer’s league with a constructive program can be an admirable aid to any form of government but a league which will attempt nothing but a cry of no taxes means nothing less than a disruption of business and more taxes in the end.

Finally, the following is from the Brooklyn Daily Eagle (Brooklyn, New York) of 29^th August 1925:

A FAMILIAR COMPLAINT.

The saving of souls by law is not new in the history of reform. The complaint of Premier Ferguson of Ontario, made to a visiting committee of Congressmen, is familiar though. “Ontario temperance organizations,” he said, “went in too much for politics and not enough for temperance.” We, who live in the shadow of organization censorship of this, that and the other thing, know how it feels.
It was not so long ago that the dispatches from Washington and from Albany made it sound as though the Anti-Saloon League were running the country and the State. Ominous threats come from blue-law* societies concerning what is going to happen in the future. It has always been so. Once a little group feels its power it goes forth brandishing the shining sword of righteousness and promising to lop off political heads with all the sinister threats of the old back-room boys who used to pull the strings and make Mayors, Governors and even Presidents dance. The true crusader is willing to hold up international affairs, important economic developments and even wars to free captive canaries or whatever it is his reforming neurosis settles upon. That is why politicians fear him. And he likes the feel of his power.

* blue law: in colonial New England, a strict law motivated by religious belief, particularly one preventing entertainment or leisure activities on a Sunday; later, in extended use, any law considered severely puritanical or restrictive, especially one forbidding secular activities such as shopping on Sunday