"The Yiddish word meaning "pray" is "daven," which ultimately comes from the same Latin root as the English word "divine" and emphasizes the One to whom prayer is directed."
N.B.: Yiddish comes from medieval German. German is not a Romance language (i.e., not a Latin derivative). So, this root word would have had to be a word that was imported into German.
Prayers and Blessings is a worthwhile read, covering the general topic of prayer.
"Jewish Services" at wikipedia derives "daven" differently:
Daven is the originally exclusively Eastern Yiddish verb meaning "pray"; it is widely used by Ashkenazic Orthodox Jews. In Yinglish, this has become the Anglicised davening. The origin of the word is obscure, but is thought by some to have come from Middle French divin (short for office divin, Divine service) and by others to be derived from a Slavic word meaning "to give" (Russian: давать, davat') . Others claim that it originates from an Aramaic word, "de'avoohon" or "d'avinun", meaning "of their/our forefathers", as the three prayers are said to have been invented by Abraham, Isaac and Jacob. In Western Yiddish, the term for "pray" is oren, a word with clear roots in Romance languages—compare Spanish and Portuguese orar and Latin orare.
Given the ways in which the various dialects of Yiddish evolved, incorporating words from the vernacular where the speakers lived, this explanation seems quite reasonable. If "daven" came from the middle French "divin," because French is a Romance language, deriving it from the same Latin root as "divine" remains a possibility.
Rabbi Simche Sherer (a dayan of the Beit Din, Manchester, England) writes:
According to the Admor (Rebbe) of Vishnitz, the explanation of word "daven" is as follows.
It is not German, Latin, French or Hebrew. It is an original Yiddish word.
In the Beis Hamikdosh, the people did not just sing and pray, the main service involved sacrifices. Until the arrival of Moshiach, we do not have this facility and thus daven instead.
There are 3 Tefillos; Shachris (to which Mussaf was added, hence it's name), Mincha and Maariv. Each were devised by our forefathers, Avrohom, Yitzchok and Yaakov. The word "daven" is a derivative of D'AVINU, [meaning] from our fathers. We are now left only with these prayers, the custom of our forefathers. We do not generally create ad lib prayers as do peoples of other faiths.
אוסיף שבספרו של ברלינר ששה חדשים באיטליה :פברואר-יולי 1873 : רשמי מסע מקהילות ישראל וספריות עבריות [עברית: ישראל א. זידמן ; הערות ותיקונים: נפתלי בן-מנחם ירושלים :מוסד הרב קוק,תש"ה 1945] כותב שהמקור הוא במילה ערבית וכבר נכתב בהערה שם שדבריו בעניין זה אינם נכונים.