In the present: just the shards and R grails. And the R grails aren't a real choice, so yea, just the eye shards.

In the not-so-distant-anymore-future: 5 cores are needed (along with other materials) to final limit break certain old advent SSR weapons.
Because advent SSRs gain FLB eligibility with their corresponding event (2nd Reprints, so there are only a handful of advent SSRs that can be FLB'd on DMM so far), you can get away with farming the cores necessary to FLB during those events, if you're confident in your ability to clear Ragnarok then.

For example, the first 2nd Reprint that'll introduce FLB for its weapon will be Phoenix in September, when the Phoenix bow gets FLB implemented. If you think that you can handle Phoenix ragnarok, then you can pick up the 5 cores for FLB then (oh, and uh, set aside 25 gold chest and 45 silver chest materials...).

Aside: All FLBs also require varying amounts of orichalcum which come from Guild Order, which is an event type that should be introduced next month. I'll leave it at that.