Is it obligatory to give up an overnight stay that was obtained through bribery, or is repentance sufficient, and how can one convince their mother of the impermissibility of bribery?

Bribery is forbidden and is a major sin. The Prophet, peace and blessings be upon him, cursed the one who gives and the one who takes a bribe. It is permissible to pay it out of necessity, such as to ward off injustice or to obtain a right. However, what you mentioned about the distance does not justify paying a bribe to obtain a forged residency permit for university housing. This combines bribery, forgery, transgression, and deception. You must sincerely repent, and among the conditions of repentance are vacating the housing or informing the officials of the truth of the matter. If they permit you to stay, then it is permissible.