Question

which would it be: "my mom never has to settle arguments among my brother, my sister and me? (or I?)"

Answer 1

me. The easiest way to know whether to use I or me is to eliminate the rest of the people from the sentence.

My mom never has to settle arguments with I.

See that does not work.

My mother never has to settle arguments with me.

Therefore,

My mother never has to settle arguments among my brother, my sister, and me.