After getting the info from her mother, you have to talk to her, in which case she'll 'invite' you to come along. Meet her near the forest entrance on Friday after that, and you should be taken to the location, through an event, and made to do a puzzle. -- If you mess up, use the red string (given to you before puzzle starts).Where does Kay hide on Fridays? Her mom says she has a secret place that she knows where she is. but I looked everywhere open but couldn't find her. Do I need to check EVERY spot in EVERY building?
Update: Sorry, I did not know that this later became a formal quest itself.
Again, to re-illiterate:
1. Get info from talking with Kay's mother (RNG based)
2. Talk to Kay afterwards
3. Now she will be near the forest entrance on Friday Mornings (until you talk to her and do the event)
4. Do the puzzle.