Is it permissible for a childless paternal aunt to designate her share of the family home to her brother, who passed away before her, given her refusal to bequeath it to anyone else, or can she make a bequest of one-third of it to them instead?

Your aunt's writing dedicating her share to your father is considered a will to him, and it is null and void because your father passed away before her. A will to an heir is a matter of dispute among jurists, oscillating between being disliked (makruh) and prohibited (haram). She is permitted to bequeath to someone who is not an heir, provided it does not exceed one-third of her wealth if she is wealthy. For a poor person, it is preferable to leave their wealth to their heirs. Furthermore, severing ties of kinship does not permit depriving an heir of their inheritance.