There are several, conflicting, stories about Minilla’s background; and, I don't think there is a definitive consensus on where he actually came from.
Some stories contradict one another, or are at least ambiguous. Others were created as the franchise developed and expanded. That, and there's much apocrypha, too!
Even his name is somewhat problematic.
Minilla: That is the anglicized version used for the U.S. release and Minira is the Japanese version ('mini' and 'Gojira'). Mina and Milla are also used in dubbed versions of the film. In scripts running up to the filming, but prior to the naming contest created by Toho for the film, he was called Kodomo Gojira (child godzilla) or Gojira Junia (godzilla junior). In the shooting script he's named Kojira (childzilla).
Godzilla Junior: As referred to in Godzilla vs. Destoroyah (1995). Although Minilla's name sometimes is also the name of a different Kojira in the Heisei series.
This is not at all comprehensive, BTW. On top of this, there are variations of his name due to multiple versions across different media, over a long timeline, including language changes based upon translations (by many people) of dubbing and sub-titles!?!
Including a time when he shows up for no apparent reason: Godzilla: Final Wars (2004).
What is generally accepted is that he—and it is a 'he' (Ryffe, 1998, p. 141)—is a godzillasaurus. Minilla as seen in Son of Godzilla (1967), as noted in the question above, was born prematurely due to the Kamacuras attack.
In Son of Godzilla he is adopted and trained by Godzilla. However, there is no indication that he is related directly (biologically) to Godzilla… only that he is adopted.
In, for example, Go! Greenman (1973 TV series), he is a blood-thirsty creation of Tonchiki, a servant of Maoh.
As the franchise developed and expanded across media (film, TV, etc.) two versions appear: the goofy version we know and love/hate and a darker version as seen in the Go! Greenman TV series.
In the end, director Jun Fukuda said Minilla was simply created to increase production value and draw a broader, more diverse, audience.
In this way, then, although there is a somewhat coherent idea for Minilla, there is no clear or definitive origin story.
Minilla's origin story is, quite literally, apocrypha.
Resources (the first three are—IMO—the most informative):
- Japan's Favorite Mon-Star: The Unauthorized Biography of "The Big G", Steve Ryffe (1998)
- "The Complete Godzilla Chronology 1954-2004" by August Ragone and Bob Eggleton (2014; Famous Monsters of Filmland. No. 274.).
- "Twenty-five Years with Godzilla" by Ed Godziszewski (1979; Fangoria. No. 1)
- All Toho Monsters Pictorial Book, published by Yosensha (2014)
- Toho Special Effects Movie Complete Works, published by Village Books (2012)
- Encyclopedia of Godzilla (Mechagodzilla Edition), published by Gakken (1993)
- The Toho Studios Story: A History and Complete Filmography, by Stuart Galbraith (2008)
- The Official Godzilla Compendium, J.D. Lees and Marc Cerasini (1998)