The story everyone tells
The legend is irresistible and it is repeated everywhere: Square was months from collapse in 1987, Hironobu Sakaguchi had one shot left, and he named the game Final Fantasy because it would be the company's final anything if it flopped. Then it sold, the company survived, and the most successful role-playing franchise in history is named after the moment it nearly did not exist. It has the shape of a perfect anecdote — a title that means one thing at the time and something ironic in retrospect.
Every element of the setting is true. Square was faced with the possibility of bankruptcy by 1987. Final Fantasy did succeed, and the company's survival is genuinely attributable to it. The problem is the causal link in the middle: Sakaguchi has said the name did not come from the company's finances. The story is not a fabrication so much as a correct set of facts wired together in the wrong order.
What Sakaguchi actually says
By his account, the team wanted an abbreviation that worked in the Latin alphabet, in the manner of Dragon Quest, which was becoming known as DQ. They arrived at the letters first: FF. The name that originally filled those initials was Fighting Fantasy — which ran into trademark difficulty, the term already belonging to the British gamebook series by Steve Jackson and Ian Livingstone. So the team needed a different word beginning with F, and went with Final somewhat reluctantly, as a last resort to preserve the abbreviation they had already settled on.
This is a considerably less romantic origin: a branding decision, a trademark conflict, and a compromise. But there is a residue of the legend that survives, and Sakaguchi has confirmed it — "Final" was chosen partly because he was planning to leave the games industry, and this was going to be his last game. The finality was real. It was personal rather than corporate.
Why the wrong version wins
The bankruptcy version persists because it is a better story, and because it is assembled entirely from true parts. Square was nearly bankrupt. The game was called Final Fantasy. The game saved the company. A reader who knows those three facts will construct the causal link themselves without anyone having to assert it, which is how this kind of legend propagates — nobody has to lie, they simply have to place the facts adjacently and let the reader do the work.
It also flatters everyone involved. A designer naming his game after his employer's imminent death is a more heroic figure than a team working backwards from a two-letter abbreviation after a trademark search went badly. And the actual story — that "Final" meant Sakaguchi's own retirement — is arguably stranger and more human than the legend, since it means the title was a private statement about one person's career that happened to get printed on ten million boxes.
The finality that did not happen
The irony that survives the correction is the good one. Sakaguchi named the game Final because he intended to leave the industry afterwards; instead it succeeded, and he stayed for sixteen more years, shepherding a franchise whose name announces an ending that never came. Final Fantasy X-2 was his last credited project at Square, and he left in 2003 — the finality arriving a decade and a half late and by an entirely different route.
The company's fortunes tracked him more closely than anyone expected. Nobuo Uematsu has said that Square "suddenly collapsed" after Sakaguchi's departure and that the situation there was awful once he had quit — a striking assessment of a man whose original plan was to leave before any of it started. The naming legend gets the drama right and the mechanism wrong, which is the usual failure mode of a good anecdote. What is true is that a game called Final Fantasy was named by someone who meant it about himself, was wrong about that, and spent the next sixteen years being wrong about it in public.