Is there sin in refusing to give the اثر (trace) of ablution to a brother suffering from the evil eye, and is there any Shar'i (Islamic legal) evidence for taking the اثر specifically from family members?

The evil eye is real. If the person who cast the evil eye is known, he should be asked to perform ghusl (ritual bath), and the water from his ghusl should be poured over the afflicted person. The evil eye can even come from a righteous and loving person. The one who cast the evil eye must promptly take action to remove its effect. Scholars have differed on the obligation of ghusl for the one who cast the evil eye, but the most correct view is that it is obligatory. There is no harm in the afflicted person asking his brothers and children to perform ghusl if there is evidence that the evil eye originated from them. As for the ruqyah practitioner asking for ablution water to be brought to him, if it is for the purpose of treating the afflicted person with it, then it is permissible.