Are mosque activities, such as dedicating a day for gathering to perform Qiyam al-Layl after the Isha prayer to encourage one another, considered Bid‘ah (innovation), knowing that specifying a day is for organizational purposes and not out of belief in its virtue?

There is no harm in designating one night of the week for Qiyam al-Layl (night vigil prayer), because most scholars have permitted supererogatory prayers, both in congregation and individually. Ibn Qudamah stated in Al-Mughni: "It is permissible to perform supererogatory prayers in congregation and individually, because the Prophet, peace be upon him, did both." However, it is important not to designate a specific night for this, to avoid falling into the prohibition mentioned in that regard, as the Prophet, peace be upon him, said: "Do not single out the night of Jumu'ah (Friday) for Qiyam among other nights." This prohibition includes other nights if they are specifically designated.