There's really not much to add here...
What's the connection between Eliyahu and the Brit Milah? We have a kiseh shel eliyahu (Chair of Eliyahu) and we mention his name when the infant is placed on the chair.
From Explaining the Customs of Bris Milah (relevant excerpt):
WHY IS THERE A CHAIR FOR ELIYAHU?
According to the Midrash, Eliyahu Hanavi attends every bris. Before Eliyahu rose to heaven and assumed the role of an angel, he was the prophet responsible for admonishing the wicked monarchs Achav and Izevel. Eliyahu was a zealot for Hashem’s honor (Melachim 1:19:10, 14) and accused Bnei Yisrael of abrogating Bris Milah. As a response, Hashem decreed that Eliyahu would be present at every bris to see that the Jews indeed fulfill bris milah. Chazal therefore instituted the custom that there should be a seat of honor for Eliyahu at every bris (Pirkei D’Rabbi Eliezer, Chapter 29; Zohar 93a). Eliyahu thus came to be called the "Angel of the Covenant," since he attends and attests to every bris. Therefore, the chair that the baby is placed upon before the bris is referred to as Kisay shel Eliyahu.