StarFox began life as a tech demo, a "look at what we can do" demonstration of technology. Argonaut had put together a little shooter level with a rudimentary Arwing and a few towers and enemies to shoot down. Nintendo took a look at it and, being impressed, they commissioned Argonaut to build a game around the technology... but not using the shooter level or any of its elements.
Enter Shigeru Miyamoto who, according to the legend, saw potential in the little level and envisioned a more complex gaming environment with wingmen, bosses, and other classic Miyamoto touches. Nintendo was well aware of their top designer's success record and commissioned the Arwing/shooter game.
While Argonaut did handle the programming and production of the final StarFox game, Miyamoto designed many of the elements including Fox McCloud himself and his wingmen Falco Lombardi, Peppy Hare, and Slippy Toad. He also designed Andross and put together the basic plot of the game. He also came up with a name for the game... StarWolf. Well, that was one of the proposed names. Before Fox McCloud was a fox he was a wolf, a sheep (StarSheep), and a whole host of other concepts before Miyamoto settled on a fox for his lead character (The StarWolf concept would be used later in StarFox 64).
While StarFox was successful and beloved by the gaming masses, Shigeru Miyamoto was not entirely happy. He had envisioned so much more in the StarFox world that, several years later, the game would be remade as StarFox 64 and would feature everything the designer wanted in the original game.
References:
The StarFox 64 Players Guide
A long-discarded copy of Electronic Gaming Monthly with an interview with Argonaut's programmers
Footnote: I tried to research more on this topic on Google, but was unable to find any information beyond what I already knew. Searching for historical StarFox material on Google is quite a challenge because there's an entire elaborate fanbase devoted to creating their own StarFox fanfiction.