Why do we call praying Davening? What is the source of this word? Where did this originate?
It originates from the Yiddish language where:
דאַוונען = to pray
So just do a little yiddish -> english transliteration magic and you get:
Couple of other notes:
It was originally called doynen in old Yiddish, before it changed to davnen (source)
As for the etymology: many say it is unknown.
But... you can find a bunch of supposed ones...
Here are two quick ones from here:
Balashon says "the theory that makes the most sense to me, and seems to be the most widely accepted," is that it comes from Latin divinus, which means "divine". See those two links for more info, the first about daven and the second about divinus.
Per this link http://hebrewbooks.org/pdfpager.aspx?req=3350&hilite=d4efb5ed-7aba-4e47-9119-42505177041b&st=%d7%a9%d7%99%d7%a8+%d7%94%d7%a9%d7%99%d7%a8%d7%99%d7%9d+%d7%a2%d7%a8%d7%91+%d7%a9%d7%91%d7%aa&pgnum=231 it comes from Aramaic.
Artscroll Shemoneh Esrei in the introduction says as follows. "The Yiddsish word for prayer "Davenen" derives from the Aramaic term "D'avunan" - from our fathers; because prayer isthe precious legacy handed down to us from our forefathers, the Avos".
In a lecture R' Adin Steinsaltz mentioned that he thinks it derives from the word daf (page). A person who has a transcendetal experience is truly praying, but one who merely turns the pages is dafenen.
I ordinarily like to link to the lectures I cite, but this one I heard in person on Shabbat, so there is no recording.