Adding to u/Great_Bacca 's link to u/eighthgear 's answer:

Most likely no one person came up with the name. Rather it emerged naturally. George R. Stewart wrote about this in Names on the Land:

Who first may have happened to say United States of America can never be known. A hundred people may have done so independently, because the elements of the name were all in common use. Fittingly, however, the first recorded use was from the pen of Thomas Jefferson in the Declaration of Independence: "WE, THEREFORE, The Representatives of the United States of America…"

(although as eighthgear points out, Jefferson did not capitalize "united")

Stewart says the name's origins can be traced back to before the Revolutionary War. At first, he says, people simply spoke of the "united colonies". This soon became the United Colonies of America or the United Colonies of North America. When war broke out commissions in the new army were at first given with one of these names. However, the problematic nature of the word "colonies" during a war for independence quickly became clear, and "states" was a natural and obvious alternative.

However, no name was officially declared and there were variations. The treaty of alliance with France used the name United States of North America, but this longer form never gained popularity. The abbreviation U.S.A. was used from very early in the war, as was the short form "United States".

At first this was an excellent name, but its flaws soon became apparent. Stewart again:

At the time [late 1770s] and for the immediate ends, the name United States of America was excellent. It represented the least possible break with tradition, and thus did not unduly frighten the conservative. It raised no question about subordinating the individual states. It was even an argument of the legitimacy of the Revolution...

As soon, however, as the emergency of the war had grown less, the inadequacy of the name became apparent. United States of America was greatly lacking in that it supplied no good adjective or term for the inhabitants of the country. It was unwieldy, inexact, and unoriginal. Although it rolled well from the tongue of an orator, not even the sincerest patriot could manage it in a poem or song.

As far as alternative names, Columbia was "the chief rival". As for why:

Columbus had never risen as a hero during the colonial period. Great Britain had always, for political reasons, emphasized the Cabots' discovery of North America. By the time of the Revolution, however, there was no chance that Spain would extend a claim of sovereignty over New York or Philadelphia, and the Cabots were shadowy agents of a British king, unheroic in stature. The new nation began to look back toward Columbus as a kind of founding hero.

Stewart goes on to argue how much better Columbia would have been:

Columbia was a happy coinage. Virginia and Georgia had already made such names familiar. It was almost everything that the United States of America was not—short, precise, original, poetic, indivisible, and flexibly yielding good adjectives and nouns...

And how the ideal time to make a change was at the Constitutional Convention of 1787, but that during the convention:

There seems also to be some kind of unimaginative quality in statesmen which makes them think that a name should describe, even at the expense of being awkward. Of the two chief men who might have argued for a better name, Franklin (that old maker of proverbs) was no longer vigorous, and Jefferson was in France...

Even after 1787 there was "some agitation for Columbia", and other names such as Fredonia, proposed by Samuel Latham Mitchill shortly after 1800. After 1819 Columbia, or Colombia, became associated with the new country in South America and thus: "was no longer available as a national name".

Around that time Washington Irving suggested Appalachia or Alleghania. Later, another proposal was to use the first letters of United States of North America to make the name "Usona".

None of these alternatives ever gained enough traction to actually be seriously debated in Congress. Stewart makes his own opinion quite clear:

The makeshift establishment of the national name was the worst misfortune in our whole naming-history. Its too great length has consumed paper, ink, time, and energy. Its vagueness and inaccuracy have caused incalculable misunderstanding, and bad feeling. Yet the trouble has never been acute enough to occasion an amendment to the Constitution, and any official change has become less and less likely.

On similar grounds he also condemns British Commonwealth of Nations, Soyuz Sovietskikh Sotsialisticheskikh Respublik, and United Nations.

Already answered here by u/eighthgear