It's not a single record, but several.
They start appearing after you have reached the Archive.

To add some more context, Leryn is basically locked behind story progression. You will reach the Archive naturally at certain point in the story and will be prompted to unlock her without really needing to do anything special.

When she is unlockable, she will talk to you automatically about her role on the Investigation Team upon leaving your house. Afterwards, you will unlock the records "Secret Reunion", "Hidden Helper", and "Potion Development Journal" in sequential order. Watching these 3 records will unlock Leryn as a playable character.

I looked at the records and reviewed them, but Lerun remained blocked (bug probably), so I edited save to unlock her. I’ll leave it here, maybe it will be useful to someone.

Path C:\Users\"Username"\AppData\LocalLow\AlFine\ChronoArk
open in notepad your Save1.sav, 2 or 3 (depends on slot),
search for <UnlockCharacter>
add <string>Leryn</string> above </UnlockCharacter>
save file, good luck

@fellhead omg that helped so much. I already unlocked her but for some reason I couldn't gift her any items. There was a requirement for a record where you have to lose to a certain enemy but I've only ever lost either in the beginning or end of my runs so I never got it before