in canon it would be her late mother's necklace. she traded it away once in order to do 'the right thing'/make amends for her mistakes, but after getting it back later she wouldn't want to part with it again.🥺

in sail: it might be the capacity core grandioze, a memento from the game's final dungeon. in typical jrpg fashion, this was where she'd lost her mentor, older brother, and lover. very depressing. so there's a lot of feelings around both of those items lmao.