Would my mother be sinful for giving a portion of the alms money bequeathed by her father to her brother, without her father's knowledge?

As long as your mother's father did not specify a particular recipient or group of people for the charity (ṣadaqah), and he entrusted her with absolute authority in its distribution, and her brother was in need, then there is no blame upon her for giving the charity to him, and she is not considered to have transgressed by doing so.