It's widely known that the original NES Final Fantasy was heavily inspired by the pen-and-paper RPG Dungeons & Dragons (to the extent of over 90% of the monsters in FF coming straight from the pages of the Monster Manual) but I think I've found a possible pen-and-paper inspiration for FFL as well.
In the late 1970s, around the same time the first edition of Advanced D&D, TSR published an RPG titled "Gamma World" set on a post-apocalyptic Earth. The inhabitants of the Gamma World (and thus the player characters in the game) come in three types: "Pure" humans, mutant humans, and sentient (mutated) animals of various species. Mutant characters roll dice against a "mutation table" to determine what superhuman abilities they possess--including mental powers like ESP and telekinesis, fire and electrical attacks, and a power called "life leech" which drains HP from every being in the vicinity, friend and foe alike.
Although there were experience points and levels in Gamma World, compared to D&D they were rather unimportant--in particular, characters do not get more HP when they gain levels, and most of a character's power comes from items ("lost technology" from the pre-apocalypse society) and mutations rather than from experience points.
The resemblance to FFL is striking--in particular, the "life leech" mutation is identical to a certain Mutant ability in FFL. I don't think Gamma World was ever published in Japan, but I've heard that Akitoshi Kawazu (game designer of both FF and FFL) used to import and play foreign RPGs, board games, and war games when he was in university, translating the rules into Japanese on his own.
It's also worth noting that a later edition of Gamma World added robots as a fourth playable character type...