What is the ruling on masturbating approximately weekly during the Athan, and what must one who has done so and wishes to repent do?

You did well by advising your friend to desist from his shameful act, and continue to encourage him to repent. Be careful not to spread such news. If your friend openly commits sins, then inform him of the obligation to repent from open transgression, for the Prophet, peace and blessings be upon him, said: "All of my Ummah are safe, except for the open transgressors..." Committing disobedience at the time of the Adhan (definition) indicates heedlessness and weakness of faith. It is incumbent upon a Muslim to answer the call of Allah and to approach the prayer, which forbids indecency and evil.