OMORI

First of all, the Pet Rock mini-game is just Rock, Paper, Scissors. It depends on percentage and which attacks your opponent might use, so good luck!
After buying Hero's Gift, interact with the Shopkeep again, and he'll offer a Pet Rock for $10. It has only 3 hearts, but that's enough to beat everyone. To get to the champion, you first have to defeat four random people who have rock symbols above them (indicating they want to clash). These people are located in Hobbeez itself, Gino's Pizza, Faraway Park, Charlene's House and Artist's House. If you defeat four of them (I didn't defeat Angel in Artist's House because of story reasons), interact with the Shopkeep and he'll tell you need to defeat three Pet Rock veterans, located in Gino's, Fix-It and Othermart. After defeating them, you have to defeat the Shopkeep as well. He'll hint towards someone who is the real Pet Rock champion. It's the old lady who sells fish in Othermart. She is very tough, with 9 hearts and her attacks are random, with 33% on every attack.

The Omori Wiki has estimates of chance of specific attacks during the fights, that being:

Katie (Hobbeez, 3 hearts) - Rock (60%), Paper (40%), Scissors (0%)
Rai (Gino's, 3 hearts) - Rock (25%), Paper (50%), Scissors (25%)
Pretty Boy (Faraway Park, 3 hearts) - Rock (0%), Paper (60%), Scissors (40%)
Tucker (Charlene's House, 3 hearts) - Rock (40%), Paper (0%), Scissors (60%)
Angel (Artist's House, 6 hearts) - Rock (0%), Paper (0%), Scissors (100%)
Veterans:
Gino (Gino's, 4 hearts) - Rock (33%), Paper (33%), Scissors (33%)
Curtsey (Othermart, 5 hearts) - Rock (50%), Paper (25%), Scissors (25%)
Mando (Fix-It, 5 hearts) - Rock (25%), Paper (25%), Scissors (50%)
Pedro (Hobbeez, 5 hearts) - Rock (25%), Paper (50%), Scissors (25%)
Ayee (Othermart, 9 hearts) - Rock (33%), Paper (33%), Scissors (33%)

Bonus: if you want true randomness and luck, use a random number generator. Maybe you will get lucky!