The following is probably not directly related to the history of this name but the word "Ora" is part of the Latin Hail Mary prayer. The line is "Sancta Maria, Mater Dei, Ora pro nobis peccatoribus" meaning "Holy Mary, Mother of God, pray for us sinners". I believe the word "Ora" is the Latin imperative meaning "pray".Perhaps this name's popularity in the late 19th century and early 20th century USA was due to its sound. If you check the statistics, Cora, Nora, Dora, Mora, Zora and Aura were also very popular names at the time.
― Anonymous User 6/24/2012
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Ora is a popular name in Israel and the meaning is "light" "bright" from Or. [noted -ed]