Is it permissible for a sister to breastfeed her brother when their mother is ill, and what is the Shar'i (Islamic legal) evidence for that?

Yes, it is permissible for a sister to nurse her brother. Consequently, he becomes a son to her and to her husband who caused the milk to be produced. Based on this, it becomes forbidden for him to marry the daughters of this man, or his sisters, or his maternal aunts, or his paternal aunts. This is based on the Almighty's saying: "Forbidden to you are your mothers, and your daughters" [An-Nisa: 23], and the Prophet (peace be upon him) said: "What is forbidden by lineage is also forbidden by suckling."