The custom to daven for one's needs by a bris when the baby cries is well known and has been considered a segula. The idea, that I have heard, is that Eliyahu Hanavi is present at every bris and when the baby cries it arouses God's mercy and in Eliyahu's merit, or with him as a messenger, one's prayers will be answered.

Someone just asked Rav Benzion Mutzafi what the source is for this custom.

According to a report on Kikar, Rav Mutzafi responded that there is no source. Rav Mutzafi said the entire thing is made up - it is not a custom and not a segula - there is no source for it. Even more so, the opposite is actually true - those present at a bris merit to receive an extar neshama (similar to on Shabbos, I guess), but when the baby cries and feels the moment of pain, all the guests lose their neshama and all that remains is the nefesh. In order to get the neshama back, Rav Mutzafi says, the people at the bris make the bracha on besamim, similar to what happens on Motzei Shabbos.

I must say, that does not sound better and more reasonable to me.
1. Even worse - at Ashkenazi circumcisions they do not normally pass around spices or herbs to bless and smell. Do the guests at an ashkenazi bris not get their neshama back until the coming Shabbos?
2. Why should the baby's cries make all the people present lose their neshamas?
3. It sounds like those present lose both neshamas. Do people really not have a neshama until the spices come out or is it only the extra one? At the end of the bris, does the extra neshama leave on its own? What does the extra neshama do while it is with us?