Mother of High Priest
Rav Elazar Abuchatzeira explained the strange explanation that the mother of the Kohein Hagadol food and clothing. This was meant to deter the family of the one stuck in the City of Refuge from praying that her son would die. This was because the person was released upon the death of the High Priest.
He further asked how such prayers could actually be so effective that Hashem would actually allow the Kohein Hagadol to die. To this, he answered that when one prays with a broken heart, he is capable of shaking the Heavens to the point that such prayers could be answered.
This is where the gifts of the mother come in. Since the family will have feelings of gratitude, Hakarat Hatov, to her, they will no longer be able to pray with a broken heart that the High Priest should die.
In this way, the gifts of the mother of the כהן הגדול, could save her son’s life.