This is the first time I had this problem. When I did automatic weights I had to redo the hands since in I had to clean up the vertices and the fingers were stuck together. One the left hand I had no problems but the right hand I duplicated and mirrored and simply attached the hand. I attached the bones and parented them, but when I move the bones it is moving other fingers not attached to the current bone. I checked the vertex groups and they are .
I would try selecting all the vertex points you want to represent the right hand, and then press Ctrl + i to invert the selection to the rest of the mesh, and press Remove to remove them from the right-hand vertex group, while the right-hand vertex group is selected and highlighted of course.