How can I atone for my sin after stealing my father's watch and accusing another person of it, which led to his imprisonment, especially since I don't know the whereabouts or name of this man?

Theft is a clear sin, and your silence regarding the imprisonment and fining of this person is a great sin. What is required of you for repentance is: repentance to Allah Almighty, and fining your father the value of the watch unless he absolves you of it. Then, inform him to return what he took of money. Strive to find his whereabouts to return the money and seek his absolution. If you are unable, then give in charity on his behalf what he was fined.