The Final Fantasy III remake has an influence that goes understated. With more drastic changes compared to the GBA ports and being the first Final Fantasy game that Tomoya Asano directed, eventually leading to the Bravely Default series (which still has some infleunces from III), the FFIII remake is definitely a significant aspect of the series, at least on some level.
But, as the unused text strings hint, it was probably not that smooth of a ride.
As is probably well known by this point, Final Fantasy III was intended to get a Wonderswan port much like its older brothers. The Wonderswan ports had a significant impact on later ports of Final Fantasy I and II, becoming the basis for those ports with the redone music and graphical style.
So when III didn't get that port...
The reasoning why appears to be simple: III is too big for its own good. Final Fantasy III is one of the biggest NES games, and for good reason, that it completely filled the cartridge. Not only did SE need to port it to another console, they also had to touch up the graphics, redo the music, tweak some mechanics, and overall give it the same flourish that the other two games got. In an interview with Eurogamer, Hiromichi Tanaka states that it wasn't just cartridge size that held the game back: It was time and manpower too.
...Didn't exist, but it was considered.
Prior to landing on the DS, Square-Enix considered porting the game to the PS2. This wasn't far-fetched, as Dragon Quest V had been remade for the platform by ArtePizarra and Matrix Software, the latter of whom would also end up involved in the FFIII remake. In any case, as luck would have it, Nintendo approached S-E with their new portable platform, the Nintendo DS, and asked if they would support the machine. The decision was soon made to put the Final Fantasy III remake on the DS.
OTher things that I learned about FF3R's development:
Final Fantasy XI shares quite a bit of DNA with III. This is no surprise: Hiromichi Tanaka and Koichii Ishii both worked on the original Final Fantasy III and Final Fantasy XI was essentially them creating their own ideal Final Fantasy. However, the remake interestingly features a list of every weapon from XI. In this interview, Tanaka mentions that he originally intended to reuse models from XI, but eventually had to redo all of them. It's possible that the weapon list is because the intention was to port over most of the weapons, at least until they could be scaled back to what the player would end up using.
The Final Fantasy III remake is a hard game. The enemies hit hard and the bosses hit twice as hard (a statement that is more literal than one would think). When the topic comes up, the assumption is that the difficulty is a result of scaling gone wrong: SE couldn't get more than three enemies on-screen at once, so they jacked up all of the stats to compensate for the player being in less danger.
That's not quite accurate.
An interview at the back of the official strategy guide reveals that it wasn't that simple: Kazuhiko Aoki and Hiroaki Yabuta reveal that the goal was to create a system that would force the player to make tense choices or risk getting party-wiped. The devs estimated how close they wanted the party to get to death in each dungeon and tweaked them accordingly.
The Cloud of Darkness was apparently also tweaked right up until the deadline.
Final Fantasy III has a simple rather multiplayer mode. The Mognet system allows players to share messages with each other and unlock the Legendary Smith, Onion Knight and Iron Giant sidequests. However, it wasn't always intended to be this simple.
According to the interview mentioned above, a Player Vs Player mode was conceptualized. The specifics were not mentioned, but it seems that Luneth, Arc, Refia and Ingus could pick a job and fight against each other. It's said that there were actually quite a few new abilities created specifically for this mode. However, as the team thought about it more and more, they came to realize that they felt uncomfortable with the idea of Luneth and his friends fighting each other. Thus, the mode was scrapped and the abilities along with it.