The word POTUS is an acronym from the initial letters of President of the United States.

Like SCOTUS, an acronym from the initial letters of Supreme Court of the United States, POTUS was originally an abbreviation used in the telegraphic code created in 1879 by the American journalist, telegraph operator and businessman Walter P. Phillips (1846-1920). This was explained in Telegraphic Shorthand, published in The Railroad Telegrapher: A Semimonthly Magazine Published in the Interest of the Railroad Telegraphers of North America (The Telegraphers Publishing Co, Vinton, Iowa) of 1^st May 1894:

The typewriter and the Phillips code have revolutionized the methods of telegraphy as the typesetting machine has changed the composing room of a newspaper. An expert operator can now take at least twice as many words in a day as he could five years ago. He has adapted the typewriter to his use to give himself speed and to meet the increased facility, the sending operator has utilized the Phillips code so that it is literally a telegraphic shorthand. Before the typewriter was made a part of the equipment of the operator he was obliged to write out the press reports in longhand just as they appeared in the newspaper and this involved an amount of close work which few constitutions could stand. Now with much less fatigue the operator copies a much longer report.
The Phillips code is the invention of Walter P. Phillips, secretary of the United Press and has come into general use on all telegraph lines. It is a system of abbreviations as puzzling as the word signs of phonography. The receiving operator must know his code well enough to recognize the word intended instantly he gets the abbreviation.
Not long ago there appeared in a morning paper a statement that some prominent man might possibly be nominated for Scotus. Everyone who read the paragraph must have been puzzled to know where Scotus was. This was a case where the operator had failed to catch the abbreviation and of course it was all Greek to the editor.
Scotus is the abbreviation of the Phillips code for Supreme Court of the United States and is a good example of the condensation which comes with progress in this important department of a newspaper.
The sentences given below will serve to show ordinary uses of the Phillips code.
T chw xt afa caud gx.
The chairman committed suicide and the affair caused great excitement.
T potus tdy sent to t sa a kmn anog 5 apn md ft Scotus & t td ws brot or fm las y.
The President of the United States today sent to the Senate a communication announcing that the appropriation made for the Supreme Court of the United States and the Treasury department was brought over from last year.
Those experts in the system of telegraphic shorthand can take at the rate of seventy-five words a minute. It is no uncommon thing for the sending operator to put 3,000 words on the wire in sixty minutes by making an average of fifty words a minute, a speed which would be impossible to take a few years ago.

The occurrence of POTUS in this text antedates the earliest instance recorded in the Oxford English Dictionary (3^rd edition – 2006), which is from 1895. I have found even earlier occurrences in The Philadelphia Inquirer (Philadelphia, Pennsylvania) of 17^th January 1894; they may represent erroneous retention of the telegraphic abbreviation—as mentioned about SCOTUS in the previous text:

                                                                                IN THE SENATE.
By Associated Press.
Washington, Jan. 16.—The President’s message transmitting the Hawaiian correspondence sent into the executive session yesterday was laid before the Senate to-day and read. Mr. Hoar (Rep., Mass.) […] proceeded to discuss the President’s message. He said: “It will be remembered that the potus, thinking it would be a shame to expose the founders of the provisional government to the cruelties of a semi-barbarous Queen, stipulated that they should be granted complete amnesty. When that suggestion was made to her she said she thought the parties should be beheaded, their property confiscated and their family and children driven from the realm. […]
“There is but one point of sympathy between the potus and the Queen of Hawaii. That was a purpose on the part of each recklessly to disregard the Constitution of their own countries (laughter), the only difference being that the potus disregards the Constitution recklessly without provocation, and the Queen of Hawaii in this instance proposes to disregard it only before the great temptation of a throne and on the urgent solicitation of our President.”