Is there a specific supplication for the deceased, and is there any merit to a mother's death occurring in the months of Sha'ban or Ramadan?

Supplicating for the deceased is permissible with any formula of mercy and seeking forgiveness. The best way is through the supplications authentically reported from the Prophet, peace be upon him, such as: "O Allah, forgive him, have mercy on him, pardon him, and grant him well-being. Make his abode honorable and widen his entrance. Wash him with water, snow, and hail, and cleanse him from sins as a white garment is cleansed from dirt. Give him a home better than his home, a family better than his family, and a spouse better than his spouse. Protect him from the trial of the grave and the torment of the Fire." There is no evidence for the superiority of dying in Sha'ban or Ramadan.