In the original Final Fantasy there is only one way to get bonus stats: to level up. You can level at most 49 times, and can gain at most 1 of each stat per level so there is a very real hard cap in terms of how good a given character (say, a thief) can be.
The stats you gain on level up aren't really as random as you might think though. It turns out each level has a fixed set of stats you're guaranteed to gain (level 2 was a strength, an agility, and a luck for example) and then you have a chance (supposedly 50%) to also get the other stats. So theoretically I could save before every level gain and reset if I didn't get every stat, but that will take a long time. The question is, which stats should I actually scum for? What do the stats do? I did some digging and have a rough idea:
Strength - Every 2 strength increases the damage done per swing by an average of 1.5. I think I'll have 10 attacks at max level with fast up, so every strength is 7.5 damage per attack. This seems pretty good.
Agility - Your base evade is 40 + your agility. No one really knows what the evade formula is, but it seems reasonable to me that the attack rolls a number between 0-255 and if it's under your evade you don't get hit. If that is true, then an agility is worth 0.4% dodge. I'm not sure if the plan for killing bosses is going to involve the ruse shirt or not (probably yes) but if not then this seems good. Otherwise pretty irrelevant.
Intelligence - As best I can tell this stat has no game effect.
Endurance - When you level up you gain max health based on a formula. Either you gain 1 + (End/4) or you gain 20 + d6 + (End/4). So gaining an endurance at level 10 will likely get me 10 max health when I hit the level cap. The early couple endurance are probably worth scumming for, but later ones don't seem too important. I'm better off scumming for a better d6 roll on my big health levels, really.
Luck - Apparently when you're trying to run away the game rolls a number between 0 and (16+x) and then compares that to your luck. If the roll is less than your luck, you run away. But the game is bugged, and whatever they meant x to be it's something else entirely. If the runner is in the first slot then x is based on the status of the character in the third slot. In my case that status will always be dead, so x is 1. As such, if I have at least 18 luck I'm guaranteed to run away. I'm level 6 now and have 19 luck, so unless they fixed the bug in the Playstation version I'm guaranteed to run away.
So it seems like Int and Luck are worthless so I might as well scum for Str and Agi. Maybe End too if I feel like it at the time but I won't worry if I miss some.