This sentence should be "She OWES me a big favor" not "owns". The word "owe" means to "have an obligation to pay or repay for something received." e.g. If you lent money to someone, that person has to return the money back to you, which means that he "owes" you the money you gave him. The sentence "She owes you a big favor" means that you probably did a big favor to that person in the past so she has to do you a big favor too now in return. I hope this helps!

correct is " she owes me a big favor"

It means that one day you expect her to return you favour/help.