Chidushei Maharz Binga, a talmid muvhik of the Maharil, writes that he saw people pour a glass of wine and call it Kos Shel Eliyahu and that a possible reason for this is since we are expecting him to come and he will need wine for the Arba Kosos.
Rav Moshe Chagiz writes that the reason for Kos Shel Eliyahu was similar to the reason we prepare a special chair for Eliyahu Hanavi at a bris milah. Just as Eliyahu Hanavi witnesses that the bris is performed, here too, on Pesach, Eliyahu Hanavi is to witness that the Korban Pesach is done properly. The Korban Pesach is dependent on milah, since the halacha is that only someone with a bris milah can eat the Korban Pesach.