Final limit breaks. The weapons that get final limit breaks made available for them are the draconic eye exchange weapons, SSR kamihime weapons, union event weapons (when they occur for the 3rd time?), and original advent SSRs when their 2nd Reprints occur.

For non-whales, draconic eye weapons and SSR kamihime weapons are mostly out of the picture. So your FLB options are mainly coming down to union event weapons and the advent SSRs. Union events occur more frequently than 2nd Reprints, so the list of weapons that you'd consider to FLB will be predominantly union event weapons for quite a while. Consider that on DMM right now, the list of 2nd reprints to date is Phoenix->Apocalypse->Rahab->Typhon->Vritra. On the other hand, all 12 union event weapons (2 per element) have FLB available. Oh, yea, union events get a 2nd SSR weapon starting on their 3rd go around or so?

What's the point of FLB?
- raise the level cap from 125 to 150
- raise the skill level cap from 20 to 30
- either improve a currently existing skill or add a second one
- raise your soul's burst multiplier by one step?

For the Pride staff specifically, FLB gives it Ascension (medium). Ascension's +heal; that is, it'll increase both healing power (small amount but who cares) and the upper limit (the important thing here). (medium) means that it'll raise heals' upper limit by (7 + skill_level)%. So when you take a Pride staff and FLB it, that lv 20 Ascension (medium) raises healing cap by 27%. Then you take it to lv 30 and now that's +37% heal.
Ascension stacks additively. So let's say you FLB a second Pride staff and raise it to skill lv 30. Now your light team is rocking +74% heal. Almost the same effect as 50 grails in +heal in union events.
Oh, and lv 30 Pride skill still gives +15% to assault at minimum.

Oh, and the costs of FLB don't include needing another copy of the weapon. They require farming a bunch of other stuff.