The idea of Hekate having an Owl sitting upon her shoulder appears to have originated in 2008 from a personal description of Hekate Trivia described by Jade Sol Luna. Not only is it an opinion but includes his own invocation and chant to be used for calling her on a specific day of the week. There is an accompanying photo included in the book though no indication if it is copyrighted or public domain.
Jade Sol Luna:
“Hecate Trivia wears a midnight blue dress and has long white hair. She holds a broom in her right hand and a noose in her left hand. She is wild in personality and is accompanied by two black phantoms. An owl sits on her shoulder and she is often found in the forest at night and is sometimes accompanied by Hermes. She has a terrific laugh and sometimes wears a veil. Her invocation is: Zabarbathouch. Her chant is Ego Hecate Trivia Laudo. (for Wednesday)”
~Hecate Death, Transition and Spiritual Mastery page 69 (1st edition) ISBN 978-1438248523 A Create Space Publication (2008); page 113 (2nd edition) ISBN 978-1442184510, India Research Press / Tara Press International (2012)
What appears to have occurred though is that apart from the idea of the Owl sitting upon her shoulder everyone has dropped both the source reference and the rest of his UPG. What started out as the author’s personal thoughts and opinions has crossed over into the realm of implied fact.
Here is the image from the book. The image in the book is in greyscale while this image is in color One thing appears certain, to me anyway, that is not an owl on her shoulder in this image.